Problem 75 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$\log _{4} x=-3$$

Step-by-Step Solution

Answer

The equivalent exponential form of the equation is \(4^{-3} = x\). The solution for \(x\) is \(x = \frac{1}{64}\).

Step 1: Transform Logarithmic Equation to Exponential Form

Given the equation \(\log_{4}x = -3\), you can use the rule that \(\log_b a = c \) is equivalent to \( b^c = a \). By applying this rule, the equation \(\log _{4} x=-3\) is equivalent to \(4^{-3} = x\).

Step 2: Solve for \(x\)

To find the value of \(x\), you can simplify \(4^{-3}\). This is equivalent to \(\frac{1}{4^3}\), which simplifies further to \(\frac{1}{64}\). Therefore, \(x = \frac{1}{64}\).