Exponential and Logarithmic Functions

Solve Exponential Equation

Solve 2^x = 32 by recognizing 32 = 2^5, giving x = 5. Verify the solution by substitution.

Example: 2^x = 32. Since 32 = 2^5, the solution is x = 5. Verify: 2^5 = 32. ✓

Example

Rewrite the right side as a matching power and solve for the exponent.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

2x=322 ^{x}= 32

Step 2 — Rewrite right side

Rewrite 32 as 2^5.

25=322 ^{ \hl{5} }= \hl{32}

Step 3 — Take log base b

Take log base 2 of both sides.

x=log⁡2(32)x=\log_{ \hl{2} }( \hl{32} )

Step 4 — Solution

Matching powers gives x = 5.

x=5x= \hl{5}

Step 5 — Verify

Check: 2^5 = 32.

25=322 ^{ 5 }= 32
solve-exponential-equation To solve b^x = c, express c as a power of b: if c = b^k then x = k. Verify by substituting back: b^k must equal c.