The Laws of Logarithms: Product, Quotient and Power Rules

Q: Why does log(MN) = log M + log N?

Because logs are exponents. If M = b^p and N = b^q, then MN = b^p+q, so the log of the product is p + q.

Q: How do logs help solve equations like 5^x = 40?

Taking logs of both sides and using the power law gives x log 5 = log 40, which is an ordinary equation in x.

The product, quotient and power laws for logs, why each one follows from an exponent law, how the power law solves exponential equations, and the mistakes that look like laws but are not.

3 min read Supports 3 lessons

Logarithms are exponents, so they obey the same rules as exponents, translated. Multiplying powers adds their exponents, so the log of a product is the sum of the logs.

The three laws

Product

\log_b(MN) = \log_b M + \log_b N

Quotient

\log_b\!\left(\frac{M}{N}\right) = \log_b M - \log_b N

Power

\log_b(M^k) = k\log_b M

Why the product law works

$M = b^p, \qquad N = b^q$
Write $M$ and $N$ as powers of $b$ . Then $\log_b M = p$ and $\log_b N = q$ .

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The quotient law comes from dividing powers in the same way, and the power law from raising a power to a power.

Using the laws

Worked example

Simplify $\log_2 40 - \log_2 5$ .

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Quotient law.
$\log_2 \frac{40}{5} = \log_2 8$

Worked example

Write $2\log 3 + \log 4$ as a single logarithm.

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Power law first.
$\log 3^2 + \log 4 = \log 9 + \log 4$

Solving exponential equations

The power law is what makes logs so useful: it brings an exponent down to ground level, where ordinary algebra can reach it.

$5^x = 40$
The unknown is in the exponent.

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Tip

If both sides can be written with the same base, you do not need logs at all. $3^{x + 1} = 81 = 3^4$ , so $x + 1 = 4$ and $x = 3$ .

Change of base

The example above shows a general fact: any log can be computed with whichever log your calculator has.

Change of base

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

What is not a law

Watch out

$\log(a + b) \neq \log a + \log b$ . There is no rule for the log of a sum.

Watch out

$\frac{\log a}{\log b} \neq \log\frac{a}{b}$ . A quotient of logs is a change of base, not the quotient law.

Watch out

$(\log x)^2 \neq 2\log x$ . The power law only applies when the power is inside the log.

Common questions

Why does log(MN) = log M + log N?: Because logs are exponents. If $M = b^p$ and $N = b^q$ , then $MN = b^{p+q}$ , so the log of the product is $p + q$ .
How do logs help solve equations like 5^x = 40?: Taking logs of both sides and using the power law gives $x\log 5 = \log 40$ , which is an ordinary equation in $x$ .

Practise laws of logarithms

The Laws of Logarithms lesson has worked examples and unlimited practice that adapts to you.

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Common questions

Why does log(MN) = log M + log N?

Because logs are exponents. If $M = b^p$ and $N = b^q$ , then $MN = b^{p+q}$ , so the log of the product is $p + q$ .

How do logs help solve equations like 5^x = 40?

Taking logs of both sides and using the power law gives $x\log 5 = \log 40$ , which is an ordinary equation in $x$ .