Change of Base Formula

Why we need change of base

Most calculators only have $\ln$ (natural log) and $\log_{10}$ buttons — no $\log_5$ button. The change of base formula lets you evaluate any log using the buttons you have:

$\log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}$

Both fractions give the same answer — use whichever logarithm your calculator has.

Step-by-step: Evaluate $\log_5 17$

Step 1 — Apply the formula using natural log:

$\log_5 17 = \frac{\ln 17}{\ln 5}$

Step 2 — Compute each part on your calculator:

$\ln 17 = 2.8332$ and $\ln 5 = 1.6094$

Step 3 — Divide:

$\log_5 17 = \frac{2.8332}{1.6094} = 1.7604$

Step 4 — Check: Does $5^{1.7604} = 17$? $5^{1.7604} \approx 17.00$ ✓

Why the formula works

If $\log_5 17 = y$, then $5^y = 17$. Take $\ln$ of both sides: $y \ln 5 = \ln 17$. Solve: $y = \ln 17 / \ln 5$.

Try it in the visualization

Enter any base $b$ and argument $x$. The formula is applied step by step using both $\ln$ and $\log_{10}$. The result is verified by computing $b^{\text{result}}$.