Change of Base Formula (Converting bases)

You can convert a logarithm from one base to another using the change of base formula: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$ for any base $k > 0$.

For example, to compute $\log_2(100)$ using base $10$: $\log_2(100) = \frac{\log_{10}(100)}{\log_{10}(2)} = \frac{2}{0.301} \approx 6.64$.

This formula shows why different bases only differ by a constant factor. The ratio $\frac{1}{\log_k(b)}$ is constant, so $O(\log_2 n) = O(\log_{10} n)$ in Big-O notation.