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Chapter 12: Problem 76

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

Short Answer

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

Step by step solution

01

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 \(64^{\frac{2}{3}} = x\).
02

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 \((\sqrt[3]{64})^{2} = x\)
03

Solve for x

Solving for \(x\), we get \(4^2 = x\)
04

Final Result

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

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$$\text { Solve: } x-2(3 x-2)>2 x-3$$
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