Which base should I change to?

Choose the base that makes the numbers simplest, usually the smallest base in the question. Base 9 and base 3 both become base 3, because 9 = 3².

Can I use my calculator's log button?

Yes. The change of base rule works with any base, so log(a) / log(b) with base 10 or ln gives the same answer.

How to Solve Logarithms with Different Bases

Short answer

How do you solve logarithms with different bases?

Use the change of base rule, log_b(a) = log(a) / log(b), to rewrite every log in the same base. Then combine them with the log laws and solve. For example log₃x + log₉x = 6 becomes 1.5 log₃x = 6, so x = 81.

A logarithm answers "what power?": $lo g_{2} 8 = 3$ because $2^{3} = 8$ . When a question mixes bases, you cannot combine the logs until they share one.

The change of base rule

lo g_{b} a = \frac{lo g _{c} a}{lo g _{c} b}

for any base $c$ . A useful special case: $lo g_{4} 8 = \frac{lo g _{2} 8}{lo g _{2} 4} = \frac{3}{2}$ .

Example 1: two bases in one equation

Solve $lo g_{3} x + lo g_{9} x = 6$ .

Rewrite $lo g_{9} x$ in base 3. Since $lo g_{3} 9 = 2$ :

lo g_{9} x = \frac{lo g _{3} x}{lo g _{3} 9} = \frac{lo g _{3} x}{2}

Now both terms share a base:

lo g_{3} x + \frac{1}{2} lo g_{3} x = 6 ⟹ \frac{3}{2} lo g_{3} x = 6 ⟹ lo g_{3} x = 4 ⟹ x = 3^{4} = 81

Check: $lo g_{3} 81 = 4$ and $lo g_{9} 81 = 2$ , and $4 + 2 = 6$.

Example 2: an exponential equation

Solve $2^{x} = 7$ , giving $x$ to 3 significant figures.

Take logs of both sides, then divide:

x = lo g_{2} 7 = \frac{lo g 7}{lo g 2} = 2.807 \dots \approx 2.81

Mistakes that cost marks

Adding logs with different bases as if they matched. Change base first.
Writing $lo g_{9} x = 2 lo g_{3} x$ . It is half, because $9 = 3^2$ sits in the denominator.
Rounding early. Keep the full value until the last line.

How to solve logarithms with different bases