Operations on Logarithms

Logarithms turn multiplication into addition and division into subtraction. The equality property that solves equations, the product and quotient rules, and why the base must match.

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Logarithms have a remarkable property: they turn multiplication into addition and division into subtraction. That single shift is what made them a calculating tool for centuries, and it is still what makes logarithmic equations solvable today.

Note A quick reminder

\log_3(9) = 2 because 3^2 = 9 . The question is always: to what power must the base be raised to reach this number?

Every rule below requires all the logarithms involved to share the same base.

Property 1 Equality

\log_a(x) = \log_a(y) \quad \Longrightarrow \quad x = y

If two logarithms with the same base are equal, then whatever sits inside them must be equal too.

\log_5(x) = \log_5(8) gives x = 8 .
\log_5(x+4) = \log_5(y+2) gives x+4 = y+2 .

This is the property that lets you strip the logarithms off both sides of an equation and solve what remains.

Property 2 The product rule

\log_a(x \times y) = \log_a(x) + \log_a(y)

The logarithm of a product is the sum of the logarithms. Multiplication inside becomes addition outside.

\log_2(5 \times 6) = \log_2(5) + \log_2(6)

You can check the rule against a value you already know. Take \log_2(4) :

Split 4 as 2 \times 2 , so \log_2(4) = \log_2(2 \times 2) .
Apply the rule: \log_2(2) + \log_2(2) .
Each term is 1, so the total is 1 + 1 = 2 .
Directly, \log_2(4) = 2 because 2^2 = 4 — the same answer.
Property 3 The quotient rule

\log_a\!\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)

Division inside becomes subtraction outside — the natural counterpart of the product rule.

\log_2\!\left(\frac{8}{2}\right) = \log_2(8) - \log_2(2) = 3 - 1 = 2
Checking directly: \frac{8}{2} = 4 , and \log_2(4) = 2 .

The order matters here. The numerator's logarithm comes first, then the denominator's is subtracted.

Note The pattern behind the rules
multiply inside add divide inside subtract
Each operation drops down one level: multiplication becomes addition, division becomes subtraction. That is why logarithms simplify heavy arithmetic.
Example Solving with the equality property

Solve \log_3(2x - 1) = \log_3(7) .

Both sides are logarithms with base 3, so the insides must be equal.
2x - 1 = 7
2x = 8 , so x = 4 .
Check: 2(4) - 1 = 7 , and the input is positive, so the solution is valid.
⟹ x = 4

Always confirm that the value inside each logarithm ends up positive. A solution that makes it zero or negative has to be rejected.

Note Mistakes to avoid
Writing \log(x+y) = \log(x) + \log(y) — the product rule applies to multiplication, not addition.
Combining logarithms with different bases; every rule here needs a shared base.
Reversing the quotient rule and subtracting the numerator from the denominator.
Turning \log(x) \times \log(y) into \log(xy) — that is not what the rule says.
Accepting a solution that makes the inside of a logarithm zero or negative.
Summary
  1. All these rules require the logarithms to share the same base.
  2. Equality: if log_a(x) = log_a(y), then x = y.
  3. Product: log_a(xy) = log_a(x) + log_a(y) — multiplication becomes addition.
  4. Quotient: log_a(x/y) = log_a(x) − log_a(y) — division becomes subtraction.
  5. Always check that the value inside every logarithm stays positive.
231. Arithmetic Sequences and Series
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