Year 12 Exponentials and logarithms: Exponential equations

Exponential equations are easiest when both sides can be rewritten using the same base. Once the bases match, the exponents must be equal.

Try base rewriting first. Logarithms are the backup plan when a common base is not practical.

Main method

Isolate the exponential expression if needed.
Rewrite both sides in terms of a common base.
Equate the exponents and solve.

Worked example

Problem: Solve

9^{x} = 27

Rewrite both numbers in base 3: $9 = 3^{2}$ and $27 = 3^{3}$ .
Substitute: ${(3^{2})}^{x} = 3^{3}$ .
Simplify the left side: $3^{2 x} = 3^{3}$ .
Equal bases mean equal exponents, so $2 x = 3$ .
Solve: $x = 3 / 2$ .

Answer:

x = 3 / 2

When common bases do not appear immediately

Not every exponential equation is generous enough to show matching bases from the start. Sometimes you first isolate the power, and sometimes no useful common base exists at all. In those cases, logarithms are not a last resort because you are stuck. They are the standard tool for moving the unknown out of the exponent in a controlled way.

Worked example 2

Problem: Solve

3^{2 x + 1} = 20

A neat common-base rewrite is not possible, so take logs of both sides: $\log (3^{2 x + 1}) = \log (20)$ .
Use the power rule: $(2 x + 1) \log 3 = \log 20$ .
Divide by $\log 3$ : $2 x + 1 = \log 20 / \log 3$ .
Rearrange: $2 x = \log 20 / \log 3 - 1$ .
So $x = (\log 20 / \log 3 - 1) / 2$ .
Calculator value: $x = 0.864 . . .$ .

Answer:

x = 0.864

to 3 decimal places.

Common mistakes

Equating exponents before the bases are actually equal.
Forgetting to isolate the exponential term before taking logs.
Dropping brackets when the exponent has more than one term.
Using rounded intermediate values too early.

Revision focus

Exponential and logarithmic questions become much easier when you classify the task before starting. Ask whether you are evaluating a logarithm exactly, simplifying with log laws, solving an exponential equation, solving a logarithmic equation, or modelling growth and decay. Each of those tasks has a standard pattern, and confusion usually comes from mixing those patterns together rather than from the algebra itself.

Another high-value habit is to reverse-check the answer. If a logarithm equals a certain number, convert it back into exponential form and see whether the statement is true. If a model gives a time or population value, ask whether the rounding and units make sense in context. Domain restrictions, positive log arguments, and sensible growth factors are not side details in this topic. They are part of what makes the answer mathematically valid.

A quick final check for this topic is to ask whether the form of the answer matches the form of the question. Exact evaluations should usually stay exact. Solving questions should respect domain restrictions. Modelling questions should include sensible units, rounding, and interpretation. Those checks are short, but they are exactly the checks that stop a nearly correct solution from becoming an incorrect one.

Practice links

Equations involving exponents