Expand logarithms with steps

What this page covers

• The product rule, $\log(ab) = \log a + \log b$, and the quotient rule, $\log\frac{a}{b} = \log a - \log b$.
• The power rule, including roots written as fractional powers: $\log\sqrt{a} = \frac{1}{2}\log a$.
• Writing the log of a number in terms of given logs, such as $\log_b 72$ from $\log_b 2$ and $\log_b 3$.
• Not covered here: combining logs into one (use Condense logarithms calculator) and changing the base (use Change of base calculator).

How to enter the problem

• Type the logarithm with its argument in brackets, for example log_2(8x) or ln(e^3 sqrt(x) / 4).
• Scan it. Fit the one problem in the frame.
• Choose a photo. Several problems in one photo are solved as a list.
• Then tap Solve.

Worked examples

Common mistakes

• Splitting the log of a sum: $\ln(x^2 + 1)$ is not $\ln x^2 + \ln 1$. Only products, quotients and powers expand.
• Confusing the power of the argument with the power of the log: $\log_3 (x^2 + 1)^4 = 4\log_3(x^2 + 1)$, but $\left(\log_3(x^2 + 1)\right)^4$ is something else.
• Treating a root as a root of the log: $\ln\sqrt{x} = \frac{1}{2}\ln x$, not $\sqrt{\ln x}$.
• Reversing a quotient: $\ln\frac{a}{b} = \ln a - \ln b$, not $\ln b - \ln a$.

Checks, assumptions and limits

• Condense the expanded answer with the same rules; the original logarithm must come back.
• Compare values at one point. At $x = 4$, $\ln\frac{2e^3}{4} \approx 2.307$, and $3 + \frac{1}{2}\ln 4 - \ln 4 \approx 2.307$.
• The rules need positive arguments, so an expansion holds where every argument is positive, such as $x > 0$ for $\ln\sqrt{x}$.
• MathBuddy can make mistakes. Double check important steps.

Related tasks

• Condense logarithms into one logarithm
• Log base 2 calculator
• Factor an expression
• All algebra tasks
• All math tasks