Logarithm Change of Base

The logarithm change of base formula allows us to express a logarithm in one base in terms of logarithms of another base. This is useful because most calculators only support logarithms with base 10 (\(\log\), common logarithm) and base \(e\) (\(\ln\), natural logarithm). The formula is:

\[ \log_b(x) = \frac{\log_c(x)}{\log_c(b)} \]

where:

\( \log_b(x) \) is the logarithm of \( x \) with base \( b \),
\( \log_c(x) \) is the logarithm of \( x \) with the new base \( c \),
\( \log_c(b) \) is the logarithm of \( b \) with the new base \( c \).

Derivation of the Formula

\[ y = \log_b(x) \]\[ b^y = x \]
\[ \log_c(b^y) = \log_c(x) \]
\[ y \cdot \log_c(b) = \log_c(x) \]
\[ y = \frac{\log_c(x)}{\log_c(b)} \]
\[ \log_b(x) = \frac{\log_c(x)}{\log_c(b)} \]

Example Calculation

Let’s say we need to compute \( \log_3(100) \) using base 10 logarithms:

\[ \log_3(100) = \frac{\log_{10}(100)}{\log_{10}(3)} \]

Using a calculator:

\( \log_{10}(100) = 2 \) (since \( 10^2 = 100 \))
\( \log_{10}(3) \approx 0.4771 \)

Therefore:

\[ \log_3(100) = \frac{2}{0.4771} \approx 4.19 \]

Why is this Useful?

It allows us to compute logarithms of any base using a calculator that only supports base 10 or base \( e \).
It simplifies complex logarithm expressions when working with different bases.
It is used in computational applications where only certain logarithm functions are available.

Latex

\log_b(x) = \frac{\log_c(x)}{\log_c(b)}