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Solve the following logarithm problem for the positive solution for
x.

log_(x)64=(6)/(5)
Answer:

Solve the following logarithm problem for the positive solution for $x$ . $\newline$ $\log _{x} 64=\frac{6}{5}$ $\newline$ Answer:

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Quotient property of logarithms

Full solution

Q. Solve the following logarithm problem for the positive solution for

x

.

\newline

\log _{x} 64=\frac{6}{5}

\newline

Answer:

Understand the logarithmic equation: Understand the logarithmic equation. $\newline$ The given logarithmic equation is $\log_x(64) = \frac{6}{5}$ . This means that $x$ raised to the power of $\frac{6}{5}$ equals $64$ .
Convert to exponential form: Convert the logarithmic form to exponential form. $\newline$ Using the definition of a logarithm, we can rewrite the equation in its exponential form: $x^{\frac{6}{5}} = 64$ .
Find the $5$ th root: Find the $5$ th root of both sides. $\newline$ To isolate $x^6$ , we take the $5$ th root of both sides of the equation: $(x^{(6/5)})^{(5/6)} = 64^{(5/6)}$ .
Simplify the equation: Simplify the equation. $\newline$ The left side simplifies to $x$ , and the right side simplifies to the $5$ th root of $64$ raised to the $6$ th power: $x = (64^{5/6})$ .
Calculate the $5$ th root of $64$ : Calculate the $5$ th root of $64$ . $\newline$ The $5$ th root of $64$ is $2$ because $2^5 = 32$ and $4^5 = 1024$ , so $64$ must have a $5$ th root between $2$ and $4$ . Since $2^6 = 64$ , we know that $2$ is the $5$ th root of $64$ .
Raise to the $6$ th power: Raise the $5$ th root of $64$ to the $6$ th power. $\newline$ Now we raise $2$ to the $6$ th power: $2^6 = 64$ .
Write final answer: Write down the final answer. $\newline$ Since $x = 64^{\frac{5}{6}}$ and we have found that $64^{\frac{5}{6}} = 2^6 = 64$ , the positive solution for $x$ is $2$ .

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