Problem 5 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$2^{2 x-1}=32$$

Step-by-Step Solution

Answer

The solution to the equation \( 2^{2x -1} = 32 \) is \( x = 3 \).

Step 1: Express Right Hand Side as Power of 2

The first step is to express 32 as a power of 2. We know that \( 32 = 2^5 \) , so the equation becomes \( 2^{2x-1} = 2^5 \).

Step 2: Set Exponents Equal

In step 2, once the bases are the same, we can equate the exponents. The equation becomes \( 2x - 1 = 5 \).

Step 3: Solve for \( x \)

Finally, in step 3, solve the equation for \( x \). First, add 1 to both sides to get \( 2x = 6 \). Then divide both sides by 2 to obtain \( x = 3 \).