A slide rule converts multiplication into addition. That's exactly what logarithms do, the product law turns $\log(xy)$ into $\log x + \log y$, the quotient law turns division into subtraction, and the power law brings exponents down to the front. By the end of this lesson you'll manipulate logarithmic expressions with the same confidence as your index laws.
Today's hook, Astronomers measure star brightness on a logarithmic scale. When they say one star is "2 magnitudes brighter" than another, they mean it's $10^{0.8} \approx 6.3$ times brighter, not $2$ times. Without the log laws, every comparison would require computing large exponents. With them, it collapses to arithmetic.
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Orient and recall
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Worksheets
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
Without a calculator: is $\log_2 8 + \log_2 4$ bigger, smaller, or equal to $\log_2 32$? Justify your gut answer in one line.
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The two moves
Work through the core explanation before applying it.
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The two moves
+5 XP to read
There are only three log laws in this whole lesson, and each one mirrors a law of exponents you already know. Lock them in and every simplification problem becomes mechanical.
The key insight: multiplication becomes addition, division becomes subtraction, and powers move out front. Each law is a direct consequence of the corresponding index law.
$$\log_a(x^n) = n\log_a x$$
Match the operation
Multiplication becomes addition, division becomes subtraction, powers become multiplication.
Power law is key
The power law is the most frequently used: it brings exponents down in front of the log.
Sums inside don't expand
$\log_a(x + y)$ cannot be simplified. Log laws only apply to products, quotients, and powers.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
Product law: $\log_a(xy) = \log_a x + \log_a y$
Quotient law: $\log_a\!\left(\frac{x}{y}\right) = \log_a x - \log_a y$
Power law: $\log_a(x^n) = n\log_a x$
Understand
Concepts
Why each log law mirrors an index law
The difference between expanding and condensing expressions
Why $\log_a(x+y)$ has no further simplification
Can do
Skills
Apply product, quotient, and power laws to simplify expressions
Expand a single log into sums and differences
Condense sums and differences into a single logarithm
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Key terms
Product law$\log_a(xy) = \log_a x + \log_a y$, log of a product equals sum of logs.
Quotient law$\log_a\!\left(\tfrac{x}{y}\right) = \log_a x - \log_a y$, log of a quotient equals difference of logs.
Power law$\log_a(x^n) = n\log_a x$, the exponent moves to the front as a coefficient.
ExpandWrite a single log as a sum/difference of simpler logs using the laws.
CondenseWrite a sum/difference of logs as a single logarithm.
Domain restrictionLog laws hold only for $x, y > 0$ and base $a > 0$, $a \neq 1$.
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Why do the log laws work?
Work through the core explanation before applying it.
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Why do the log laws work?
core concept
The laws of logarithms mirror the laws of exponents because logarithms are exponents. If $m = \log_a x$ and $n = \log_a y$, then $x = a^m$ and $y = a^n$. Multiplying gives $xy = a^m \cdot a^n = a^{m+n}$, so $\log_a(xy) = m + n = \log_a x + \log_a y$. The same reasoning yields the quotient and power laws.
$$\begin{aligned} \log_a(xy) &= \log_a x + \log_a y \\ \log_a\!\left(\frac{x}{y}\right) &= \log_a x - \log_a y \\ \log_a(x^n) &= n\log_a x \end{aligned}$$
These hold for $x, y > 0$ and $a > 0$, $a \neq 1$.
These laws are essential. They allow us to expand, condense, and simplify logarithmic expressions, skills required for solving exponential equations, differentiating logarithmic functions, and evaluating logarithms with any base. Every algebraic manipulation of logs this year relies on exactly these three rules.
Pause, copy the product law ($\log_a(xy) = \log_a x + \log_a y$) and quotient law ($\log_a(x/y) = \log_a x - \log_a y$) with the mnemonic (multiplication becomes addition, division becomes subtraction) into your book.
Did you get this? True or false: the product law states that $\log_a x \cdot \log_a y = \log_a(xy)$.
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
Quotient law: difference of logs = log of quotient.
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$= \log_3 27 = \log_3(3^3) = 3$
$\frac{54}{2} = 27 = 3^3$, so $\log_3 27 = 3$.
PROBLEM 3 · ALL THREE LAWS
Express $\log_a\!\left(\dfrac{x^2\sqrt{y}}{z^3}\right)$ in terms of $\log_a x$, $\log_a y$, and $\log_a z$.
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$= \log_a(x^2\sqrt{y}) - \log_a(z^3)$
Quotient law first.
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$= \log_a(x^2) + \log_a(y^{1/2}) - \log_a(z^3)$
Product law on the numerator.
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$= 2\log_a x + \tfrac{1}{2}\log_a y - 3\log_a z$
Power law on each term. Note $\sqrt{y} = y^{1/2}$.
Quick check: Which expression equals $\log_5 2 + \log_5 3$?
Common errors · the 3 traps that cost marks
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Dodge the traps, then apply
Meet the mistakes that cost marks, then do it yourself.
Trap 01
Applying log laws to sums inside the argument
$\log_a(x + y)$ cannot be expanded using the product or quotient laws. Log laws only apply to products, quotients, and powers inside the argument. $\log_a(x + y)$ must stay as is.
Trap 02
Confusing $\log_a(x^n)$ with $(\log_a x)^n$
$(\log_a x)^n \neq n\log_a x$. The power law applies when the entire argument is raised to a power, not when the logarithm itself is raised to a power.
Trap 03
Writing $\log_a x \cdot \log_a y = \log_a(xy)$
A product of two logarithms is not the same as a log of a product. $\log_a x \cdot \log_a y$ has no simple closed form. Only $\log_a x + \log_a y = \log_a(xy)$ is valid.
Fill in the blank: $3\log_a x - \log_a y + \tfrac{1}{2}\log_a z$ condensed to a single logarithm is $\log_a\!\left(\rule{60px}{0.5px}\right)$.
Type the argument, e.g. x^3 sqrt(z) / y or equivalent.
Quick-fire practice · 5 problems
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Drill it, then lock it in
Run the quick drill and copy the summary into your book.
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Simplify $\log_5 2 + \log_5 3$.
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Simplify $\log_2 24 - \log_2 3$.
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Express $\log_a(x^3 y^2)$ in expanded form.
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Simplify $2\log_a x + 3\log_a y - \log_a z$.
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Evaluate $\log_2 8 + \log_2 4 - \log_2 2$.
Two truths, one lie: Two of these statements are correct. Which one is the lie?
Odd one out: Three of these can be simplified using a single log law. Which one cannot?
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Revisit your thinking
Earlier you compared $\log_2 8 + \log_2 4$ to $\log_2 32$. They are equal because the product law gives $\log_2 8 + \log_2 4 = \log_2(8 \times 4) = \log_2 32$. The three log laws (product, quotient, power) are direct consequences of the corresponding exponent laws and they convert multiplication problems into addition problems.
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
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Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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Short answer
Write full responses, then check them against the model answers.
