Grade 12 · MHF4U · Advanced Functions

MHF4U: Laws of Logarithms

A logarithm is an exponent. $\log_a x$ asks: what power of $a$ gives $x$ ? So $\log_2 8 = 3$ because $2^3 = 8$ . Every log law is an exponent law, read backwards.

How It Works

For $a > 0$ , $a \ne 1$ , and positive $M$ and $N$ :

Product: $\log_a(MN) = \log_a M + \log_a N$ . Multiplying powers adds exponents.

Quotient: $\log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N$ . Dividing powers subtracts them.

Power: $\log_a(M^k) = k\log_a M$ . A power of a power multiplies exponents.

Two values to know on sight: $\log_a 1 = 0$ and $\log_a a = 1$ . And for a base your calculator does not have, change of base: $\log_a x = \frac{\log x}{\log a}$ . Plain $\log$ means base 10.

You use the laws in two directions. Expanding splits one log into simple pieces; condensing packs several logs into one, which is the first step in solving most log equations. To condense, move every coefficient up as an exponent first, then combine: plus terms multiply, minus terms divide.

The traps, all of which cost marks:

$\log(M + N)$ does not split. There is no sum law.

$(\log x)^2$ is not $2\log x$ . The power law only moves an exponent on the argument.

$\frac{\log M}{\log N}$ is not $\log M - \log N$ . That is change of base, not the quotient law.

The laws need the same base on every term. Mixed bases? Change them first.

When a question gives you $\log_2 3 = p$ and asks for something else, write the new number as a product, quotient or power of numbers you know, then let the laws do the rest.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
Evaluate $\log_6 4 + \log_6 9$ .
Show solution
1. Same base, so use the product law: $\log_6 4 + \log_6 9 = \log_6 36$ .
2. $6^2 = 36$ , so $\log_6 36 = 2$ .
Answer: $2$
Easy
Evaluate $\log_3 54 - \log_3 2$ .
Show solution
1. Quotient law: $\log_3 54 - \log_3 2 = \log_3 \frac{54}{2} = \log_3 27$ .
2. $3^3 = 27$ , so the value is $3$ .
Answer: $3$
Medium
Expand $\log_3\left(\frac{9x^4}{\sqrt{y}}\right)$ , for $x, y > 0$ .
Show solution
1. Quotient law: $\log_3(9x^4) - \log_3\sqrt{y}$ .
2. Product law on the first term: $\log_3 9 + \log_3 x^4 - \log_3 y^{\frac{1}{2}}$ .
3. Power law, and $\log_3 9 = 2$ : $2 + 4\log_3 x - \frac{1}{2}\log_3 y$ .
Answer: $2 + 4\log_3 x - \frac{1}{2}\log_3 y$
Medium
Write $2\log x + \frac{1}{2}\log y - 3\log z$ as a single logarithm.
Show solution
1. Coefficients up as exponents first: $\log x^2 + \log y^{\frac{1}{2}} - \log z^3$ .
2. Plus terms multiply: $\log(x^2\sqrt{y}) - \log z^3$ .
3. Minus terms divide: $\log\left(\frac{x^2\sqrt{y}}{z^3}\right)$ .
Answer: $\log\left(\frac{x^2\sqrt{y}}{z^3}\right)$
Test level
Given $\log_2 3 = p$ and $\log_2 5 = q$ , write $\log_4 22.5$ in terms of $p$ and $q$ .
Show solution
1. The given logs are base 2, so change base: $\log_4 22.5 = \frac{\log_2 22.5}{\log_2 4} = \frac{\log_2 22.5}{2}$ .
2. Write $22.5$ with the numbers you know: $22.5 = \frac{45}{2} = \frac{3^2 \times 5}{2}$ .
3. Expand: $\log_2 22.5 = 2\log_2 3 + \log_2 5 - \log_2 2 = 2p + q - 1$ .
4. Divide by 2: $\log_4 22.5 = \frac{2p + q - 1}{2}$ .
Answer: $\log_4 22.5 = \frac{2p + q - 1}{2}$

Learn It on Martian Lab

Martian Lab teaches this in the Logarithmic & Exponential Functions lesson of its MHF4U Exponential & Logarithmic Functions unit. Watch it, then take the Exponential & Logarithmic Functions practice tests to see if it stuck.

Watch the Logarithmic & Exponential Functions lesson free

Keep Going

Free Exponential & Logarithmic Functions practice test, with solutions

More Exponential & Logarithmic Functions topics: Solving Logarithmic Equations

More MHF4U topics: Long and Synthetic Division · Vertical and Horizontal Asymptotes · Proving Trig Identities · Solving Trig Equations

Every MHF4U unit and a free sample test

Start Practicing

No credit card required to start.

Get started