Indices (Exponents) • Topic 3 of 3

Solving Exponential Equations

What are Exponential Equations?

Exponential equations are equations where the variable appears in the exponent (index). For example: $2^x = 8$, $3^{2x+1} = 27$, $5^x = 125$.

How to Solve Exponential Equations:

The key principle: If the bases are the same, the exponents must be equal.

\text{If } a^m = a^n \text{ and } a > 0, a \neq 1, \text{ then } m = n

Steps to Solve Exponential Equations:

Common Base Conversions:

Number	As a Power	Number	As a Power
4	$2^2$ or $4^1$	32	$2^5$
8	$2^3$	64	$2^6$ or $4^3$ or $8^2$
9	$3^2$	81	$3^4$ or $9^2$
16	$2^4$ or $4^2$	125	$5^3$
25	$5^2$	216	$6^3$
36	$6^2$	1000	$10^3$

When Bases Cannot Be Made Same:

If bases are different and cannot be made the same, use logarithms (advanced topic). For Grade 9, bases can usually be made the same.

Solved Examples

Worked Example

Solve a standard problem on Solving Exponential Equations.

Solution

Apply the formula/method shown in the concept section above.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.If $a^x=a^y$ (with $a\ne0,1$), then:

Explanation: Equate exponents.

Q2.$2^x=8$ gives $x=$

Explanation: $8=2^3$.

Q3.$3^x=81$ gives $x=$

Explanation: $81=3^4$.

Q4.To solve an exponential equation, write both sides with the same:

Explanation: Same base.

Number	As a Power	Number	As a Power
4	\(2^2\) or \(4^1\)	32	\(2^5\)
8	\(2^3\)	64	\(2^6\) or \(4^3\) or \(8^2\)
9	\(3^2\)	81	\(3^4\) or \(9^2\)
16	\(2^4\) or \(4^2\)	125	\(5^3\)
25	\(5^2\)	216	\(6^3\)
36	\(6^2\)	1000	\(10^3\)