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

log_(64)x=(3)/(2)
Answer:

Solve the following logarithm problem for the positive solution for $x$ . $\newline$ $\log _{64} x=\frac{3}{2}$ $\newline$ 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 _{64} x=\frac{3}{2}

\newline

Answer:

Understand the logarithmic equation: Understand the logarithmic equation. $\newline$ The given logarithmic equation is $\log_{64}(x) = \frac{3}{2}$ . This means that $64$ raised to the power of $\frac{3}{2}$ should equal $x$ .
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: $64^{\frac{3}{2}} = x$ .
Calculate $64^{\frac{3}{2}}$ : Calculate the value of $64^{\frac{3}{2}}$ . Since $64$ is $2$ raised to the $6$ th power ( $2^6$ ), we can write $64^{\frac{3}{2}}$ as $(2^6)^{\frac{3}{2}}$ . Using the power of a power rule $(a^{m*n} = (a^m)^n)$ , we get $(2^6)^{\frac{3}{2}} = 2^{6*(\frac{3}{2})} = 2^9$ .
Calculate $2^9$ : Calculate $2^9$ . $2^9$ is $2$ multiplied by itself $9$ times, which equals $512$ .
Write down the solution: Write down the solution. $\newline$ Since $64^{\frac{3}{2}} = 2^9 = 512$ , the value of $x$ is $512$ .

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