Exponential equations

Solving exponential equations by writing both sides with the same base, including fractional and negative solutions.

Same base, same index

Example 1●

Solve $3^{x - 1} = 27$.

$3^{x - 1} = 3^3$

$x - 1 = 3$, so $x = 4$.

Example 2●●●

Solve $4^x = 8$.

$(2^2)^x = 2^3$, so $2^{2x} = 2^3$.

$2x = 3$, so $x = \tfrac{3}{2}$.

Example 3●●●

Solve $9^x = \dfrac{1}{3}$.

$3^{2x} = 3^{-1}$

$x = -\tfrac{1}{2}$

Flashcards

$2^x = 32$

$x = 5$

$dfrac{1}{8}$ as a power of 2

$2^{-3}$

$sqrt{3}$ as a power of 3

$3^{\frac{1}{2}}$

Practice