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Type | Label | Description |
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Statement | ||
Theorem | relogexp 13001 | The natural logarithm of positive 𝐴 raised to an integer power. Property 4 of [Cohen] p. 301-302, restricted to natural logarithms and integer powers 𝑁. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (log‘(𝐴↑𝑁)) = (𝑁 · (log‘𝐴))) | ||
Theorem | relogiso 13002 | The natural logarithm function on positive reals determines an isomorphism from the positive reals onto the reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
⊢ (log ↾ ℝ+) Isom < , < (ℝ+, ℝ) | ||
Theorem | logltb 13003 | The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (log‘𝐴) < (log‘𝐵))) | ||
Theorem | logleb 13004 | Natural logarithm preserves ≤. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 ≤ 𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵))) | ||
Theorem | logrpap0b 13005 | The logarithm is apart from 0 if and only if its argument is apart from 1. (Contributed by Jim Kingdon, 3-Jul-2024.) |
⊢ (𝐴 ∈ ℝ+ → (𝐴 # 1 ↔ (log‘𝐴) # 0)) | ||
Theorem | logrpap0 13006 | The logarithm is apart from 0 if its argument is apart from 1. (Contributed by Jim Kingdon, 5-Jul-2024.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 # 1) → (log‘𝐴) # 0) | ||
Theorem | logrpap0d 13007 | Deduction form of logrpap0 13006. (Contributed by Jim Kingdon, 3-Jul-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 # 1) ⇒ ⊢ (𝜑 → (log‘𝐴) # 0) | ||
Theorem | rplogcl 13008 | Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 21-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (log‘𝐴) ∈ ℝ+) | ||
Theorem | logge0 13009 | The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ (log‘𝐴)) | ||
Theorem | logdivlti 13010 | The log𝑥 / 𝑥 function is strictly decreasing on the reals greater than e. (Contributed by Mario Carneiro, 14-Mar-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐴 < 𝐵) → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)) | ||
Theorem | relogcld 13011 | Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (log‘𝐴) ∈ ℝ) | ||
Theorem | reeflogd 13012 | Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (exp‘(log‘𝐴)) = 𝐴) | ||
Theorem | relogmuld 13013 | The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (log‘(𝐴 · 𝐵)) = ((log‘𝐴) + (log‘𝐵))) | ||
Theorem | relogdivd 13014 | The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (log‘(𝐴 / 𝐵)) = ((log‘𝐴) − (log‘𝐵))) | ||
Theorem | logled 13015 | Natural logarithm preserves ≤. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵))) | ||
Theorem | relogefd 13016 | Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (log‘(exp‘𝐴)) = 𝐴) | ||
Theorem | rplogcld 13017 | Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 1 < 𝐴) ⇒ ⊢ (𝜑 → (log‘𝐴) ∈ ℝ+) | ||
Theorem | logge0d 13018 | The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 1 ≤ 𝐴) ⇒ ⊢ (𝜑 → 0 ≤ (log‘𝐴)) | ||
Theorem | logge0b 13019 | The logarithm of a number is nonnegative iff the number is greater than or equal to 1. (Contributed by AV, 30-May-2020.) |
⊢ (𝐴 ∈ ℝ+ → (0 ≤ (log‘𝐴) ↔ 1 ≤ 𝐴)) | ||
Theorem | loggt0b 13020 | The logarithm of a number is positive iff the number is greater than 1. (Contributed by AV, 30-May-2020.) |
⊢ (𝐴 ∈ ℝ+ → (0 < (log‘𝐴) ↔ 1 < 𝐴)) | ||
Theorem | logle1b 13021 | The logarithm of a number is less than or equal to 1 iff the number is less than or equal to Euler's constant. (Contributed by AV, 30-May-2020.) |
⊢ (𝐴 ∈ ℝ+ → ((log‘𝐴) ≤ 1 ↔ 𝐴 ≤ e)) | ||
Theorem | loglt1b 13022 | The logarithm of a number is less than 1 iff the number is less than Euler's constant. (Contributed by AV, 30-May-2020.) |
⊢ (𝐴 ∈ ℝ+ → ((log‘𝐴) < 1 ↔ 𝐴 < e)) | ||
Theorem | rpcxpef 13023 | Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) | ||
Theorem | cxpexprp 13024 | Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℤ) → (𝐴↑𝑐𝐵) = (𝐴↑𝐵)) | ||
Theorem | cxpexpnn 13025 | Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ) → (𝐴↑𝑐𝐵) = (𝐴↑𝐵)) | ||
Theorem | logcxp 13026 | Logarithm of a complex power. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ) → (log‘(𝐴↑𝑐𝐵)) = (𝐵 · (log‘𝐴))) | ||
Theorem | rpcxp0 13027 | Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
⊢ (𝐴 ∈ ℝ+ → (𝐴↑𝑐0) = 1) | ||
Theorem | rpcxp1 13028 | Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ (𝐴 ∈ ℝ+ → (𝐴↑𝑐1) = 𝐴) | ||
Theorem | 1cxp 13029 | Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ (𝐴 ∈ ℂ → (1↑𝑐𝐴) = 1) | ||
Theorem | ecxp 13030 | Write the exponential function as an exponent to the power e. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ (𝐴 ∈ ℂ → (e↑𝑐𝐴) = (exp‘𝐴)) | ||
Theorem | rpcncxpcl 13031 | Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) ∈ ℂ) | ||
Theorem | rpcxpcl 13032 | Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ) → (𝐴↑𝑐𝐵) ∈ ℝ+) | ||
Theorem | cxpap0 13033 | Complex exponentiation is apart from zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) # 0) | ||
Theorem | rpcxpadd 13034 | Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴↑𝑐(𝐵 + 𝐶)) = ((𝐴↑𝑐𝐵) · (𝐴↑𝑐𝐶))) | ||
Theorem | rpcxpp1 13035 | Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 + 1)) = ((𝐴↑𝑐𝐵) · 𝐴)) | ||
Theorem | rpcxpneg 13036 | Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐-𝐵) = (1 / (𝐴↑𝑐𝐵))) | ||
Theorem | rpcxpsub 13037 | Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴↑𝑐(𝐵 − 𝐶)) = ((𝐴↑𝑐𝐵) / (𝐴↑𝑐𝐶))) | ||
Theorem | rpmulcxp 13038 | Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶))) | ||
Theorem | cxprec 13039 | Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴↑𝑐𝐵))) | ||
Theorem | rpdivcxp 13040 | Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) / (𝐵↑𝑐𝐶))) | ||
Theorem | cxpmul 13041 | Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ) → (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝑐𝐶)) | ||
Theorem | rpcxproot 13042 | The complex power function allows us to write n-th roots via the idiom 𝐴↑𝑐(1 / 𝑁). (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑐(1 / 𝑁))↑𝑁) = 𝐴) | ||
Theorem | abscxp 13043 | Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴↑𝑐𝐵)) = (𝐴↑𝑐(ℜ‘𝐵))) | ||
Theorem | cxplt 13044 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴↑𝑐𝐵) < (𝐴↑𝑐𝐶))) | ||
Theorem | cxple 13045 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 ≤ 𝐶 ↔ (𝐴↑𝑐𝐵) ≤ (𝐴↑𝑐𝐶))) | ||
Theorem | rpcxple2 13046 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) → (𝐴 ≤ 𝐵 ↔ (𝐴↑𝑐𝐶) ≤ (𝐵↑𝑐𝐶))) | ||
Theorem | rpcxplt2 13047 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴↑𝑐𝐶) < (𝐵↑𝑐𝐶))) | ||
Theorem | cxplt3 13048 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴↑𝑐𝐶) < (𝐴↑𝑐𝐵))) | ||
Theorem | cxple3 13049 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 ≤ 𝐶 ↔ (𝐴↑𝑐𝐶) ≤ (𝐴↑𝑐𝐵))) | ||
Theorem | rpcxpsqrt 13050 | The exponential function with exponent 1 / 2 exactly matches the square root function, and thus serves as a suitable generalization to other 𝑛-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 16-Jun-2024.) |
⊢ (𝐴 ∈ ℝ+ → (𝐴↑𝑐(1 / 2)) = (√‘𝐴)) | ||
Theorem | logsqrt 13051 | Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.) |
⊢ (𝐴 ∈ ℝ+ → (log‘(√‘𝐴)) = ((log‘𝐴) / 2)) | ||
Theorem | rpcxp0d 13052 | Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴↑𝑐0) = 1) | ||
Theorem | rpcxp1d 13053 | Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴↑𝑐1) = 𝐴) | ||
Theorem | 1cxpd 13054 | Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (1↑𝑐𝐴) = 1) | ||
Theorem | rpcncxpcld 13055 | Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴↑𝑐𝐵) ∈ ℂ) | ||
Theorem | cxpltd 13056 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 1 < 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐵 < 𝐶 ↔ (𝐴↑𝑐𝐵) < (𝐴↑𝑐𝐶))) | ||
Theorem | cxpled 13057 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 1 < 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐵 ≤ 𝐶 ↔ (𝐴↑𝑐𝐵) ≤ (𝐴↑𝑐𝐶))) | ||
Theorem | rpcxpsqrtth 13058 | Square root theorem over the complex numbers for the complex power function. Compare with resqrtth 10835. (Contributed by AV, 23-Dec-2022.) |
⊢ (𝐴 ∈ ℝ+ → ((√‘𝐴)↑𝑐2) = 𝐴) | ||
Theorem | cxprecd 13059 | Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴↑𝑐𝐵))) | ||
Theorem | rpcxpcld 13060 | Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴↑𝑐𝐵) ∈ ℝ+) | ||
Theorem | logcxpd 13061 | Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (log‘(𝐴↑𝑐𝐵)) = (𝐵 · (log‘𝐴))) | ||
Theorem | cxplt3d 13062 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐵 < 𝐶 ↔ (𝐴↑𝑐𝐶) < (𝐴↑𝑐𝐵))) | ||
Theorem | cxple3d 13063 | Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐵 ≤ 𝐶 ↔ (𝐴↑𝑐𝐶) ≤ (𝐴↑𝑐𝐵))) | ||
Theorem | cxpmuld 13064 | Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝑐𝐶)) | ||
Theorem | cxpcom 13065 | Commutative law for real exponentiation. (Contributed by AV, 29-Dec-2022.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴↑𝑐𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶)↑𝑐𝐵)) | ||
Theorem | apcxp2 13066 | Apartness and real exponentiation. (Contributed by Jim Kingdon, 10-Jul-2024.) |
⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 # 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 # 𝐶 ↔ (𝐴↑𝑐𝐵) # (𝐴↑𝑐𝐶))) | ||
Theorem | rpabscxpbnd 13067 | Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 0 < (ℜ‘𝐵)) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐴) ≤ 𝑀) ⇒ ⊢ (𝜑 → (abs‘(𝐴↑𝑐𝐵)) ≤ ((𝑀↑𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π)))) | ||
Define "log using an arbitrary base" function and then prove some of its properties. As with df-relog 12987 this is for real logarithms rather than complex logarithms. Metamath doesn't care what letters are used to represent classes. Usually classes begin with the letter "A", but here we use "B" and "X" to more clearly distinguish between "base" and "other parameter of log". There are different ways this could be defined in Metamath. The approach used here is intentionally similar to existing 2-parameter Metamath functions (operations): (𝐵 logb 𝑋) where 𝐵 is the base and 𝑋 is the argument of the logarithm function. An alternative would be to support the notational form (( logb ‘𝐵)‘𝑋); that looks a little more like traditional notation. | ||
Syntax | clogb 13068 | Extend class notation to include the logarithm generalized to an arbitrary base. |
class logb | ||
Definition | df-logb 13069* | Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as (𝐵 logb 𝑋) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". The definition will only be useful where 𝑥 is a positive real apart from one and where 𝑦 is a positive real, so the choice of (ℂ ∖ {0, 1}) and (ℂ ∖ {0}) is somewhat arbitrary (we adopt the definition used in set.mm). (Contributed by David A. Wheeler, 21-Jan-2017.) |
⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) | ||
Theorem | rplogbval 13070 | Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by Jim Kingdon, 3-Jul-2024.) |
⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 # 1 ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵))) | ||
Theorem | rplogbcl 13071 | General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.) |
⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 # 1 ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) ∈ ℝ) | ||
Theorem | rplogbid1 13072 | General logarithm is 1 when base and arg match. Property 1(a) of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by David A. Wheeler, 22-Jul-2017.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 # 1) → (𝐴 logb 𝐴) = 1) | ||
Theorem | rplogb1 13073 | The logarithm of 1 to an arbitrary base 𝐵 is 0. Property 1(b) of [Cohen4] p. 361. See log1 12995. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.) |
⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 # 1) → (𝐵 logb 1) = 0) | ||
Theorem | rpelogb 13074 | The general logarithm of a number to the base being Euler's constant is the natural logarithm of the number. Put another way, using e as the base in logb is the same as log. Definition in [Cohen4] p. 352. (Contributed by David A. Wheeler, 17-Oct-2017.) (Revised by David A. Wheeler and AV, 16-Jun-2020.) |
⊢ (𝐴 ∈ ℝ+ → (e logb 𝐴) = (log‘𝐴)) | ||
Theorem | rplogbchbase 13075 | Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.) |
⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 # 1) ∧ (𝐵 ∈ ℝ+ ∧ 𝐵 # 1) ∧ 𝑋 ∈ ℝ+) → (𝐴 logb 𝑋) = ((𝐵 logb 𝑋) / (𝐵 logb 𝐴))) | ||
Theorem | relogbval 13076 | Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵))) | ||
Theorem | relogbzcl 13077 | Closure of the general logarithm with integer base on positive reals. (Contributed by Thierry Arnoux, 27-Sep-2017.) (Proof shortened by AV, 9-Jun-2020.) |
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) ∈ ℝ) | ||
Theorem | rplogbreexp 13078 | Power law for the general logarithm for real powers: The logarithm of a positive real number to the power of a real number is equal to the product of the exponent and the logarithm of the base of the power. Property 4 of [Cohen4] p. 361. (Contributed by AV, 9-Jun-2020.) |
⊢ (((𝐵 ∈ ℝ+ ∧ 𝐵 # 1) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ) → (𝐵 logb (𝐶↑𝑐𝐸)) = (𝐸 · (𝐵 logb 𝐶))) | ||
Theorem | rplogbzexp 13079 | Power law for the general logarithm for integer powers: The logarithm of a positive real number to the power of an integer is equal to the product of the exponent and the logarithm of the base of the power. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.) |
⊢ (((𝐵 ∈ ℝ+ ∧ 𝐵 # 1) ∧ 𝐶 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐵 logb (𝐶↑𝑁)) = (𝑁 · (𝐵 logb 𝐶))) | ||
Theorem | rprelogbmul 13080 | The logarithm of the product of two positive real numbers is the sum of logarithms. Property 2 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 29-May-2020.) |
⊢ (((𝐵 ∈ ℝ+ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵 logb (𝐴 · 𝐶)) = ((𝐵 logb 𝐴) + (𝐵 logb 𝐶))) | ||
Theorem | rprelogbmulexp 13081 | The logarithm of the product of a positive real and a positive real number to the power of a real number is the sum of the logarithm of the first real number and the scaled logarithm of the second real number. (Contributed by AV, 29-May-2020.) |
⊢ (((𝐵 ∈ ℝ+ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ)) → (𝐵 logb (𝐴 · (𝐶↑𝑐𝐸))) = ((𝐵 logb 𝐴) + (𝐸 · (𝐵 logb 𝐶)))) | ||
Theorem | rprelogbdiv 13082 | The logarithm of the quotient of two positive real numbers is the difference of logarithms. Property 3 of [Cohen4] p. 361. (Contributed by AV, 29-May-2020.) |
⊢ (((𝐵 ∈ ℝ+ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵 logb (𝐴 / 𝐶)) = ((𝐵 logb 𝐴) − (𝐵 logb 𝐶))) | ||
Theorem | relogbexpap 13083 | Identity law for general logarithm: the logarithm of a power to the base is the exponent. Property 6 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.) |
⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 # 1 ∧ 𝑀 ∈ ℤ) → (𝐵 logb (𝐵↑𝑀)) = 𝑀) | ||
Theorem | nnlogbexp 13084 | Identity law for general logarithm with integer base. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.) |
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐵 logb (𝐵↑𝑀)) = 𝑀) | ||
Theorem | logbrec 13085 | Logarithm of a reciprocal changes sign. Particular case of Property 3 of [Cohen4] p. 361. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℝ+) → (𝐵 logb (1 / 𝐴)) = -(𝐵 logb 𝐴)) | ||
Theorem | logbleb 13086 | The general logarithm function is monotone/increasing. See logleb 13004. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by AV, 31-May-2020.) |
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+ ∧ 𝑌 ∈ ℝ+) → (𝑋 ≤ 𝑌 ↔ (𝐵 logb 𝑋) ≤ (𝐵 logb 𝑌))) | ||
Theorem | logblt 13087 | The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 13003. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.) |
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+ ∧ 𝑌 ∈ ℝ+) → (𝑋 < 𝑌 ↔ (𝐵 logb 𝑋) < (𝐵 logb 𝑌))) | ||
Theorem | rplogbcxp 13088 | Identity law for the general logarithm for real numbers. (Contributed by AV, 22-May-2020.) |
⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 # 1 ∧ 𝑋 ∈ ℝ) → (𝐵 logb (𝐵↑𝑐𝑋)) = 𝑋) | ||
Theorem | rpcxplogb 13089 | Identity law for the general logarithm. (Contributed by AV, 22-May-2020.) |
⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 # 1 ∧ 𝑋 ∈ ℝ+) → (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋) | ||
Theorem | relogbcxpbap 13090 | The logarithm is the inverse of the exponentiation. Observation in [Cohen4] p. 348. (Contributed by AV, 11-Jun-2020.) |
⊢ (((𝐵 ∈ ℝ+ ∧ 𝐵 # 1) ∧ 𝑋 ∈ ℝ+ ∧ 𝑌 ∈ ℝ) → ((𝐵 logb 𝑋) = 𝑌 ↔ (𝐵↑𝑐𝑌) = 𝑋)) | ||
Theorem | logbgt0b 13091 | The logarithm of a positive real number to a real base greater than 1 is positive iff the number is greater than 1. (Contributed by AV, 29-Dec-2022.) |
⊢ ((𝐴 ∈ ℝ+ ∧ (𝐵 ∈ ℝ+ ∧ 1 < 𝐵)) → (0 < (𝐵 logb 𝐴) ↔ 1 < 𝐴)) | ||
Theorem | logbgcd1irr 13092 | The logarithm of an integer greater than 1 to an integer base greater than 1 is not rational if the argument and the base are relatively prime. For example, (2 logb 9) ∈ (ℝ ∖ ℚ) (Contributed by AV, 29-Dec-2022.) |
⊢ ((𝑋 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2) ∧ (𝑋 gcd 𝐵) = 1) → (𝐵 logb 𝑋) ∈ (ℝ ∖ ℚ)) | ||
Theorem | logbgcd1irraplemexp 13093 | Lemma for logbgcd1irrap 13095. Apartness of 𝑋↑𝑁 and 𝐵↑𝑀. (Contributed by Jim Kingdon, 11-Jul-2024.) |
⊢ (𝜑 → 𝑋 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → (𝑋 gcd 𝐵) = 1) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝑋↑𝑁) # (𝐵↑𝑀)) | ||
Theorem | logbgcd1irraplemap 13094 | Lemma for logbgcd1irrap 13095. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.) |
⊢ (𝜑 → 𝑋 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → (𝑋 gcd 𝐵) = 1) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐵 logb 𝑋) # (𝑀 / 𝑁)) | ||
Theorem | logbgcd1irrap 13095 | The logarithm of an integer greater than 1 to an integer base greater than 1 is irrational (in the sense of being apart from any rational number) if the argument and the base are relatively prime. For example, (2 logb 9) # 𝑄 where 𝑄 is rational. (Contributed by AV, 29-Dec-2022.) |
⊢ (((𝑋 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) ∧ ((𝑋 gcd 𝐵) = 1 ∧ 𝑄 ∈ ℚ)) → (𝐵 logb 𝑋) # 𝑄) | ||
Theorem | 2logb9irr 13096 | Example for logbgcd1irr 13092. The logarithm of nine to base two is not rational. Also see 2logb9irrap 13102 which says that it is irrational (in the sense of being apart from any rational number). (Contributed by AV, 29-Dec-2022.) |
⊢ (2 logb 9) ∈ (ℝ ∖ ℚ) | ||
Theorem | logbprmirr 13097 | The logarithm of a prime to a different prime base is not rational. For example, (2 logb 3) ∈ (ℝ ∖ ℚ) (see 2logb3irr 13098). (Contributed by AV, 31-Dec-2022.) |
⊢ ((𝑋 ∈ ℙ ∧ 𝐵 ∈ ℙ ∧ 𝑋 ≠ 𝐵) → (𝐵 logb 𝑋) ∈ (ℝ ∖ ℚ)) | ||
Theorem | 2logb3irr 13098 | Example for logbprmirr 13097. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.) |
⊢ (2 logb 3) ∈ (ℝ ∖ ℚ) | ||
Theorem | 2logb9irrALT 13099 | Alternate proof of 2logb9irr 13096: The logarithm of nine to base two is not rational. (Contributed by AV, 31-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (2 logb 9) ∈ (ℝ ∖ ℚ) | ||
Theorem | sqrt2cxp2logb9e3 13100 | The square root of two to the power of the logarithm of nine to base two is three. (√‘2) and (2 logb 9) are not rational (see sqrt2irr0 11878 resp. 2logb9irr 13096), satisfying the statement in 2irrexpq 13101. (Contributed by AV, 29-Dec-2022.) |
⊢ ((√‘2)↑𝑐(2 logb 9)) = 3 |
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