Problem 76 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$\log _{64} x=\frac{2}{3}$$

Step-by-Step Solution

Answer

The exponential form of the equation \(\log _{64} x=\frac{2}{3}\) is \(16 = x\). Therefore, \(x = 16\).

Step 1: Convert from Logarithmic to Exponential Form

To convert the given equation from logarithmic form \(\log _{64} x=\frac{2}{3}\) to exponential form, use the base of the logarithm \(64\) and raise it to the other side of the equals sign \(\frac{2}{3}\). This is based on the rule \(b^{y} = x\) if and only if \(\log _{b} x = y\). That gives us \(64^{\frac{2}{3}} = x\).

Step 2: Calculate the Exponential Expression

Now, calculate the expression \(64^{\frac{2}{3}}\). This is the same as taking the cube root of \(64\) first and then squaring the result. This gives us \((\sqrt[3]{64})^{2} = x\)

Step 3: Solve for x

Solving for \(x\), we get \(4^2 = x\)

Step 4: Final Result

The final answer after calculating the above expression would be \(16 = x\)