Change of Base Calculator

Instantly convert a logarithm from one base to another using the change of base formula. Works with any positive base (except 1). You can even use e for natural log.

Logarithm Base Converter

Enter the original base \(b\) and the new base \(B\) (both >0 and ≠1).
Type e for Euler's number (natural base) if needed.

Result (change of base)

\( \log_b(x) = ? \)

⚡ The result means:
\( \log_{b}(x) = K \cdot \log_{B}(x) \) with \( K = \frac{1}{\log_B(b)} \).
The calculator shows \( K \) directly.

Change of Base Formula

Given \( \log_b (x) \), you can change to any base \( B \) ( \( B>0, B\neq 1 \) ) :

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

Equivalently: \(\displaystyle \log_b (x) = \left(\frac{1}{\log_B(b)}\right) \log_B(x)\)

Worked examples

Example 1

Change \( \log_2 (x) \) to natural base \( e \).

\( b = 2,\; B = e \)

\[ \log_2(x) = \frac{\log_e(x)}{\log_e(2)} = \frac{1}{\ln 2}\,\ln x\]

\(\displaystyle \frac{1}{\ln 2} \approx 1.4427\) → \(\log_2(x) \approx 1.4427\,\ln x\)

Example 2

Rewrite \( \log_4 (x) \) with base \( 2 \).

\( b = 4,\; B = 2 \)

\[ \log_4(x) = \frac{\log_2(x)}{\log_2(4)} = \frac{\log_2(x)}{2} = 0.5\,\log_2(x)\]

So \(\log_4(x) = 0.5 \log_2(x)\)

Practice activities using the calculator

Try these manually, then check with the calculator above. Enter \(b\) (original base) and \(B\) (new base).

a) \( \log_{16}(x) \), \( B = 4 \)
b) \( \log_{10}(x) \), \( B = e \)
c) \( \log_{0.1}(x) \), \( B = 10 \)
d) \( \log_{2}(x) \), \( B = 8 \)
e) \( \log_{8}(x) \), \( B = 16 \)
f) \( \log_{0.01}(x) \), \( B = 10 \)
g) \( \log_{25}(x) \), \( B = 5 \)
h) \( \log_{e}(x) \), \( B = 10 \)