Evaluate the logarithm. Round your answer to the nearest thousandth. log_(4)((1)/(19))~~

Question

Evaluate the logarithm. Round your answer to the nearest thousandth.  log_(4)((1)/(19))~~

Bytelearn · Accepted Answer

Understand Problem: Understand the problem and identify the logarithmic property to use. 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. 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 $\log_4(1)$. 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. 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. 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. 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. Rounding $-2.123$ to the nearest thousandth gives us $-2.123$.

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

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Evaluate the logarithm.\newlineRound your answer to the nearest thousandth.\newlinelog⁡4(119)≈ \log _{4}\left(\frac{1}{19}\right) \approx log4​(191​)≈

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