Logarithm of Power/General Logarithm

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 $14$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.