Problem 9 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution to the equation \(32^x = 8\) is \(x = \frac{3}{5}\)

Step 1: Express in Power of the Same Base

Rewrite both 32 and 8 as powers of 2. So, \(32^x = 8\) transforms to \((2^5)^x = 2^3\).

Step 2: Simplify

By knowing the rule of exponent which states that \((a^m)^n = a^{mn}\) we simplify to get \(2^{5x} = 2^3\).

Step 3: Equating Exponents

Having the same bases (2's) on both sides of the equation, we can set the exponents equal to each other. This gives \(5x = 3\).

Step 4: Solving for variable

Finally, solve the equation for \(x\). We do this by dividing each side by 5 to get \(x = \frac{3}{5}\).