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Evaluate the logarithm.
Round your answer to the nearest thousandth.

log_(4)((1)/(19))~~

Evaluate the logarithm. $\newline$ Round your answer to the nearest thousandth. $\newline$ $\log _{4}\left(\frac{1}{19}\right) \approx$

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Product property of logarithms

Full solution

Q. Evaluate the logarithm.

\newline

Round your answer to the nearest thousandth.

\newline

\log _{4}\left(\frac{1}{19}\right) \approx

Understand Problem: Understand the problem and identify the logarithmic property to use. $\newline$ We need to evaluate the logarithm of a fraction, which is $\log_4\left(\frac{1}{19}\right)$ . To do this, we can use the property of logarithms that states $\log_b\left(\frac{P}{Q}\right) = \log_b(P) - \log_b(Q)$ .
Apply Property: Apply the logarithmic property to the given expression. $\newline$ Using the property from Step $1$ , we can write $\log_4\left(\frac{1}{19}\right)$ as $\log_4(1) - \log_4(19)$ .
Evaluate Log( $1$ ): Evaluate $\log_4(1)$ . $\newline$ The logarithm of any number at its own base is $1$ , and since $1$ is the multiplicative identity, $\log_4(1) = 0$ .
Evaluate Log( $19$ ): Evaluate $\log_4(19)$ using a calculator. $\newline$ Since $19$ is not a power of $4$ , we need to use a calculator to find the value of $\log_4(19)$ . This is typically done by using the change of base formula: $\log_4(19) = \frac{\log(19)}{\log(4)}$ .
Perform Calculation: Perform the calculation using the change of base formula. $\newline$ Using a calculator, we find that $\log(19) \approx 1.27875$ and $\log(4) \approx 0.60206$ . Therefore, $\log_4(19) \approx \frac{1.27875}{0.60206} \approx 2.123$ .
Combine Results: Combine the results to find the final value. $\newline$ Now we combine the results from Step $3$ and Step $5$ : $0 - 2.123 = -2.123$ .
Round Final Value: Round the result to the nearest thousandth. $\newline$ Rounding $-2.123$ to the nearest thousandth gives us $-2.123$ .

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