Solving Exponential Equations

Strategy: rewrite both sides with the same base

When solving exponential equations, the key idea is: if $b^m = b^n$, then $m = n$. So rewrite both sides as powers of the same base, then set the exponents equal.

Step-by-step solution: Solve $3^{2x-1} = 81$

Step 1 — Rewrite 81 as a power of 3.

$81 = 3^4$ (since $3 \times 3 \times 3 \times 3 = 81$).

Step 2 — Now both sides have base 3:

$3^{2x - 1} = 3^4$

Step 3 — Since the bases are equal, set the exponents equal:

$2x - 1 = 4$

Step 4 — Solve the linear equation:

$2x = 5$ $x = \frac{5}{2} = 2.5$

Step 5 — Check: $3^{2(2.5) - 1} = 3^{5-1} = 3^4 = 81$ ✓

What if you can't match bases?

If both sides can't be written with the same base (e.g., $5^x = 17$), use logarithms: take $\log$ of both sides.

$x \log 5 = \log 17 \implies x = \frac{\log 17}{\log 5} \approx 1.76$

Common bases to know

$2^1 = 2$, $2^2 = 4$, $2^3 = 8$, $2^4 = 16$, $2^5 = 32$, $2^6 = 64$, $2^{10} = 1024$

$3^1 = 3$, $3^2 = 9$, $3^3 = 27$, $3^4 = 81$, $3^5 = 243$

$5^1 = 5$, $5^2 = 25$, $5^3 = 125$, $5^4 = 625$

Try it in the visualization

Adjust the base and target power. The equation is rewritten with matching bases, and the exponents are equated. The graph shows $y = b^{2x-1}$ and $y = \text{target}$ intersecting at the solution.