Logarithm of Power/General Logarithm
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Theorem
Let $x \in \R$ be a strictly positive real number.
Let $a \in \R$ be a real number such that $a > 1$.
Let $r \in \R$ be any real number.
Let $\log_a x$ be the logarithm to the base $a$ of $x$.
Then:
- $\map {\log_a} {x^r} = r \log_a x$
Proof
Let $y = r \log_a x$.
Then:
\(\ds a^y\) | \(=\) | \(\ds a^{r \log_a x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a^{\log_a x} }^r\) | Exponent Combination Laws | |||||||||||
\(\ds \) | \(=\) | \(\ds x^r\) | Definition of Logarithm base $a$ |
The result follows by taking logs base $a$ of both sides.
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Laws of Logarithms: $7.12$
- 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Computations Using Logarithms
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: $(12)$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: Exercise $14$
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