An exponential equation has the unknown in an exponent. Familiar powers such as and suggest a common base: rewrite both sides using that base, then equate their exponents. On a timed test, graph both sides in Desmos and click their intersection. Multiply the whole exponent when raising a power to a power; don't equate exponents while the bases still differ.
Hints
- Hint 1
The base is the number being raised to a power. Both and are powers of . Rewrite them that way before deciding whether the exponents can be set equal.
- Hint 2
An intersection is where two graphs share a point. Graph each side of the equation in Desmos, then click the crossing to read the input that makes them equal.
Step-by-step
Approach 1: Spot the common base, then use Desmos
- Step 1
Rewrite both bases
Since and , the given equation becomes:
The original bases differ, so setting equal to would skip the powers of .
- Step 2
Solve the rewritten equation in Desmos
Type and to graph the two sides of the rewritten equation. Click their intersection: Desmos shows . At that point the two sides are equal, so . The solution is . Choice A.
Approach 2: Match exponents by hand
- Step 1
Apply the power-of-a-power rule
Using and , multiply each outside exponent by the exponent on :
Keep the parentheses: the and each multiply an entire exponent.
- Step 2
Set the exponents equal
Both sides now have base . Powers of are equal only when their exponents are equal, so:
- Step 3
Distribute on both sides
Multiply each number outside parentheses by both terms inside:
- Step 4
Move the variable terms together
Subtract from both sides:
- Step 5
Isolate the variable term
Add to both sides; the on the right becomes :
- Step 6
Find the solution
Divide both sides by :
So the solution to the given equation is . Choice A.