Exponential Equations Worksheets

What is an exponential equation?

In an exponential equation the variable is in the exponent, as in $3^{x} = 81$. The worksheets solve them with one idea: if two powers of the same base are equal, their exponents are equal. In symbols, for a positive base $b \neq 1$, $b^{m} = b^{n}$ means $m = n$.

So the work is to rewrite both sides as powers of the same base. The power of a power rule, $(b^{m})^{n} = b^{mn}$, does most of that work. When no common base exists, logarithms are needed; every equation on these worksheets can be solved without them.

• Easy — the right side is a power of the same base (2, 3 or 5), and the answer is a whole number.
• Medium — the left base is itself a power (4, 8, 9 or 27), so the answer is a fraction.
• Hard — as Medium, but the exponent also has a coefficient, such as $9^{3x}$.

Worked examples

Easy: $3^{x} = 81$

Since $81 = 3^{4}$, the equation is $3^{x} = 3^{4}$. Answer: $x = 4$.

Medium: $8^{x} = 32$

Both 8 and 32 are powers of 2: $8 = 2^{3}$ and $32 = 2^{5}$.

Answer: $x = \tfrac{5}{3}$.

Hard: $9^{3x} = 27$

Write both sides as powers of 3: $9 = 3^{2}$ and $27 = 3^{3}$.

Answer: $x = \tfrac{1}{2}$.

Check your answers

Substitute the answer back in. A fractional exponent means a root and a power: $8^{5/3} = (\sqrt[3]{8})^{5} = 2^{5} = 32$, and $9^{3 \cdot \frac{1}{2}} = 9^{3/2} = (\sqrt{9})^{3} = 27$.

Common mistakes

• Dividing the numbers. $8^{x} = 32$ does not give $x = 32 \div 8 = 4$; check: $8^{4} = 4096$.
• Setting exponents equal too early. $8^{x} = 32$ does not mean $x = 5$. The bases must match before the exponents can be compared.
• Adding exponents instead of multiplying. $(2^{3})^{x} = 2^{3x}$, not $2^{3 + x}$.
• Forgetting the coefficient. In the hard example, $6x = 3$ gives $x = \tfrac{1}{2}$, not $x = 3$.

Sample problems

These problems come from the same generator as the worksheet above, three at each difficulty level. Press Generate for a fresh set.

Easy

1. $5^{x} = 25$
2. $5^{x} = 625$
3. $3^{x} = 3$

Medium

4. $9^{x} = 2187$
5. $8^{x} = 4$
6. $8^{x} = 16$

Hard

7. $4^{3x} = 16$
8. $27^{2x} = 27$
9. $9^{2x} = 27$

Tips for teachers and parents

• A small table of powers of 2, 3 and 5 (up to $2^{7} = 128$, $3^{5} = 243$ and $5^{5} = 3125$) makes Easy problems quick and lets students concentrate on the method.
• Medium answers are fractions on purpose: they are a natural moment to discuss what an exponent like $\tfrac{5}{3}$ means.
• These problems lead directly into logarithms, which handle equations such as $2^{x} = 10$ where no common base exists.

Related topics

• Logarithms Pre-Calculus
• Radical Expressions Algebra 2
• Order of Operations II Algebra