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log_(81)3=

$\log _{81} 3=$

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Convert between exponential and logarithmic form: all bases

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Q.

\log _{81} 3=

Identify Base: In $\log_{81}3$ , $81$ is the base. $\newline$ Rewrite $3$ as a power of $81$ . $\newline$ Since $81 = 3^4$ , we want to express $3$ as a power of $81$ . However, $3$ is not an integer power of $81$ . Instead, we can express $81$ as a power of $3$ and use the property of logarithms that allows us to invert the base and the argument (the number inside the logarithm) with a reciprocal exponent. $\newline$ $81 = 3^4$ , so $81$ $2$ .
Rewrite as Power: We found: $3 = 81^{1/4}$ $\newline$ $log_{81}3$ becomes $log_{81}81^{1/4}$ . $\newline$ Evaluate $log_{81}81^{1/4}$ . $\newline$ When logarithm base matches the argument's base, the logarithm is equal to the exponent. $\newline$ $log_{81}81^{1/4} = \frac{1}{4}$

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