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Chapter 12: Problem 9

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$32^{x}=8$$

Short Answer

Expert verified
The solution to the exponential equation \(32^x = 8\) is \(x = 3/5\).

Step by step solution

01

Expressing in terms of same base

Express 32 and 8 as powers of base 2. This can be done as: \(32 = 2^5\) and \(8 = 2^3\).
02

Substitution

Substitute these values in the original equation, it gets transformed to \( (2^5)^x = 2^3\).
03

Applying the Power of Power rule

Apply the rule of exponents that \( (a^m)^n = a^{mn}\). The equation becomes \(2^{5x} = 2^3\).
04

Equating Exponents

Now, as both sides of the equation are powers of 2, the exponents on both sides should be equal, i.e., \(5x = 3\).
05

Solving for x

By isolating x, we would just have to divide \(3\) by \(5\). The final answer will be \( x = 3/5 \).

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