Logarithms Explained with Examples

A logarithm is a question, not a new kind of number. ${\log }_{2}8=3$ simply asks: 2 to what power gives 8? The answer is $3$, because $2^3=8$. Once you read logs that way, the rules write themselves.

The three rules that solve almost everything

For a base $b$ (with $b>0$, $b\ne 1$):

1. Product rule: ${\log }_b(xy)={\log }_{b}x+{\log }_{b}y$ — multiplication inside becomes addition outside.
2. Quotient rule: ${\log }_b(x/y)={\log }_{b}x-{\log }_{b}y$ — division inside becomes subtraction outside.
3. Power rule: ${\log }_b(x^n)=n{\log }_{b}x$ — a power inside comes out front as a multiplier.

The full list, with examples, is on the logarithm rules page.

Worked examples

**Example 1: ${\log }_2(8\times 4)$. Use the product rule: ${\log }_{2}8+{\log }_{2}4=3+2=5$. Check: $8\times 4=32=2^5$. Correct.

**Example 2: ${\log }_{10}1000$. $10$ to what power gives $1000$? $10^3=1000$, so the answer is $3$.

**Example 3: $\ln e^2$. $\ln$ means log base $e$, so $\ln e^2=2$ by the power rule — the log and the exponential cancel.

Common mistakes

• Splitting a log of a sum. $\log (x+y)$ is NOT $\log x+\log y$. The product rule only works for multiplication inside.
• Forgetting the base. ${\log }_{2}8$ and ${\log }_{10}8$ are different numbers. Always check which base you are in.
• Logs of zero or negatives. $\log 0$ and $\log (-5)$ are undefined for real numbers — if your working leads there, something went wrong earlier.

Logs and exponents are inverses

Every log statement is an exponent statement in disguise: ${\log }_{b}x=y$ means exactly $b^y=x$. If a log problem ever confuses you, rewrite it as an exponent problem — and if an exponent problem confuses you, the Exponent Calculator checks the arithmetic.

Practice it

Want to check your work? Try the Log Calculator — every answer comes with steps. Then try similar problems on the practice page until reading logs as questions feels natural.