Logarithms are exponents, so the exponent rules become log rules. You will rewrite logs using the definition from Switching Between Log and Exponential Form.
Key idea
For $M, N > 0$: $$\log_b(MN) = \log_b M + \log_b N, \qquad \log_b \frac{M}{N} = \log_b M - \log_b N, \qquad \log_b M^p = p\log_b M.$$ Logs turn products into sums, quotients into differences and powers into multiples.
Why it works
Let $M = b^m$ and $N = b^n$, so $\log_b M = m$ and $\log_b N = n$. Then $MN = b^m b^n = b^{m + n}$, so $\log_b(MN) = m + n$: multiplying powers adds exponents. Dividing powers subtracts exponents, giving the quotient rule. And $M^p = (b^m)^p = b^{mp}$, so $\log_b M^p = pm$. Each log rule is an exponent rule wearing new clothes. This is how a slide rule multiplies: it adds lengths that are logs.
Worked examples
Worked example: Combining to evaluate
Evaluate $\log_2 40 - \log_2 5$.
- Quotient rule: $\log_2 \frac{40}{5} = \log_2 8$.
Answer: $3$
Worked example: Expanding
Expand $\log\left(\frac{x^3 y}{z^2}\right)$.
- Quotient, then product: $\log x^3 + \log y - \log z^2$.
- Power rule: $3\log x + \log y - 2\log z$.
Answer: $3\log x + \log y - 2\log z$
Worked example: Using known values
If $\log_b 2 = p$ and $\log_b 3 = q$, write $\log_b 18$ in terms of $p$ and $q$.
- $18 = 2 \cdot 3^2$.
- $\log_b 18 = \log_b 2 + 2\log_b 3$.
Answer: $p + 2q$
Worked example: Condensing
Write $2\ln x + \ln(x + 1) - \ln 5$ as a single logarithm.
- Power rule first: $\ln x^2 + \ln(x + 1) - \ln 5$.
- Product, then quotient: $\ln\frac{x^2(x + 1)}{5}$.
Answer: $\ln\frac{x^2(x + 1)}{5}$
Common mistake
Writing $\log(a + b) = \log a + \log b$. The product rule is about $\log(ab)$; there is no rule that splits the log of a sum. Test: $\log(10 + 10) = \log 20 \approx 1.30$, but $\log 10 + \log 10 = 2$.