Change of base formula

What the symbols mean

$y$

the logarithm you want, ${\log }_{b}x$

$b$

the base you want (positive, not 1)

$x$

the number (positive)

$\ln$

natural log (base e)

$\log$

common log (base 10)

When to use it

Use it whenever you need a log in a base your calculator does not have, such as ${\log }_{3}50$ or ${\log }_{2}1000$ (how many bits to count to 1000). Either ln or log₁₀ gives the same answer, as long as you use the same one on top and bottom.

Why it works

Let $y={\log }_{b}x$, so $b^y=x$. Take ln of both sides: $y\ln b=\ln x$, then divide by $\ln b$.

Common mistakes

• Putting the base on top: it is $\ln x/\ln b$, the number over the base.
• Cancelling the logs: $\ln 50/\ln 3$ is not $50/3$ or $\ln (50/3)$.