Change of Base

The change-of-base rule lets you rewrite a logarithm to any base $\log_a x$ in terms of logs your calculator has ($\ln$ or $\lg$). It’s what makes exponentials with awkward bases solvable.

The rule

$\log_a x = \frac{\log_b x}{\log_b a}$

In practice, use $\ln$ (or $\lg$), since those are the calculator buttons: $\log_a x = \frac{\ln x}{\ln a}$

So $\log_2 7 = \dfrac{\ln 7}{\ln 2} \approx 2.81$.

Solving exponentials with it

The rule is really the same move as solving $b^x = c$ with logs. To solve $2^x = 7$, take logs and divide: $x = \log_2 7 = \frac{\ln 7}{\ln 2} \approx 2.81$

Whether you call it “change of base” or “take logs and use the power law”, the arithmetic is identical: a division of two logs.

The key warning

$\dfrac{\ln 7}{\ln 2}$ is a division, not $\ln 7 - \ln 2$. The subtraction law would give $\ln\frac{7}{2}$, a completely different value. This mix-up is the most common error with change of base: $\frac{\ln 7}{\ln 2} \approx 2.81 \qquad \text{but} \qquad \ln 7 - \ln 2 = \ln 3.5 \approx 1.25$

They’re not the same. The change-of-base ratio divides the logs; it does not subtract them.

Why the base doesn’t matter

You can use $\ln$ or $\lg$ (or any base) in the ratio and get the same answer, because the base cancels. $\dfrac{\ln 7}{\ln 2} = \dfrac{\lg 7}{\lg 2}$. Use whichever your calculator offers.

Common mistakes

• Subtracting the logs instead of dividing them.
• Inverting the ratio ($\frac{\ln 2}{\ln 7}$).
• Rounding the two logs before dividing, losing accuracy.

Full topic context: Logs & Exponentials notes, and Laws of Logarithms.