Laws of Logarithms
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.
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.
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.
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: $\log_a(x^n) = n\log_a x$
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
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
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.
These hold for $x, y > 0$ and $a > 0$, $a \neq 1$.
Product law: $\log_a(xy) = \log_a x + \log_a y$ (multiplication → addition); Quotient law: $\log_a\!\left(\frac{x}{y}\right) = \log_a x - \log_a y$ (division → subtraction)
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
Simplify $\log_2 3 + \log_2 4$.
Simplify $\log_3 54 - \log_3 2$.
Express $\log_a\!\left(\dfrac{x^2\sqrt{y}}{z^3}\right)$ in terms of $\log_a x$, $\log_a y$, and $\log_a z$.
Quick check: Which expression equals $\log_5 2 + \log_5 3$?
Common errors · the 3 traps that cost marks
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
Simplify $\log_5 2 + \log_5 3$.
Simplify $\log_2 24 - \log_2 3$.
Express $\log_a(x^3 y^2)$ in expanded form.
Simplify $2\log_a x + 3\log_a y - \log_a z$.
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?
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.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. Simplify $\log_3 6 + \log_3 \dfrac{3}{2}$. Show working. (2 marks)
Q2. Express $\log_a\!\left(\dfrac{x^2 y}{\sqrt{z}}\right)$ in terms of $\log_a x$, $\log_a y$, and $\log_a z$. (3 marks)
Q3. Write $3\log_a x - \log_a y + \dfrac{1}{2}\log_a z$ as a single logarithm. (3 marks)
Comprehensive answers (click to reveal)
Drill 1: $\log_5 6$ · 2: $\log_2 8 = 3$ · 3: $3\log_a x + 2\log_a y$ · 4: $\log_a\!\left(\dfrac{x^2 y^3}{z}\right)$ · 5: $\log_2 16 = 4$
Q1 (2 marks): $\log_3\!\left(6 \times \tfrac{3}{2}\right)$ [0.5] $= \log_3 9 = \log_3(3^2) = 2$ [1.5].
Q2 (3 marks): $= \log_a(x^2 y) - \log_a(z^{1/2})$ [0.5] $= \log_a(x^2) + \log_a y - \log_a(z^{1/2})$ [0.5] $= 2\log_a x + \log_a y - \tfrac{1}{2}\log_a z$ [2].
Q3 (3 marks): $= \log_a(x^3) - \log_a y + \log_a(z^{1/2})$ [1] $= \log_a\!\left(\dfrac{x^3\sqrt{z}}{y}\right)$ [2].
Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
Enter the arenaClimb platforms by answering log law questions. Lighter alternative to the boss.
Mark lesson as complete
Tick when you've finished the practice and review.