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log_(4)x+log_(x)(64^(4))
Find the value of
x

$\log _{4} x+\log _{x}\left(64^{4}\right)$ $\newline$ Find the value of $x$

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Product property of logarithms

Full solution

Q.

\log _{4} x+\log _{x}\left(64^{4}\right)

\newline

Find the value of

x

Apply Logarithmic Property: Use the property of logarithms that allows the exponent to be brought to the front as a multiplier. $\newline$ $\log_{4}x + 4\log_{x}64$
Recognize Power of $4$ : Recognize that $64$ is a power of $4$ , specifically $4^3$ . $\newline$ $\log_{4}x + 4\log_{x}(4^3)$
Apply Power Rule: Apply the power rule to the second term, bringing the exponent out front. $\log_{4}x + 12\log_{x}4$
Simplify Second Term: Since $\log_{x} x = 1$ , simplify the second term. $\newline$ $\log_{4}x + 12(1)$
Combine Terms: Combine the terms. $\log_{4}x + 12$
Set Equal to Zero: Set the expression equal to zero since we're solving for $x$ . $\log_{4}x = -12$
Convert to Exponential Form: Convert the logarithmic equation to its exponential form. $\newline$ $4^{-12} = x$

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