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Chapter 12: Problem 5

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

Short Answer

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

Step by step solution

01

Identify Common Base

The given equation is \(2^{2x-1} = 32\). We observe that \(32\) can be expressed as a power of \(2\), specifically, \(32 = 2^5\).
02

Express Both Sides with Common Base

Replace \(32\) with \(2^5\), the equation becomes \(2^{2x-1} = 2^5\).
03

Equate the Exponents

With the bases being same on both sides, for this equation to hold true, the exponents should also be same. Therefore, set \(2x-1 = 5\).
04

Solve for the Variable

Solve the equation \(2x-1 = 5\) for \(x\). Add \(1\) to both sides, this gives \(2x = 6\). Then, divide by \(2\), giving \(x = 3\).

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