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

4log_(5)(413)~~

Evaluate the logarithm. $\newline$ Round your answer to the nearest thousandth. $\newline$ $4 \log _{5}(413) \approx$

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Compare linear, exponential, and quadratic growth

Full solution

Q. Evaluate the logarithm.

\newline

Round your answer to the nearest thousandth.

\newline

4 \log _{5}(413) \approx

Understanding the logarithm expression: Understand the logarithm expression. $\newline$ The expression $4\log_{5}(413)$ means $4$ times the logarithm of $413$ to the base $5$ .
Using the logarithm power rule: Use the logarithm power rule. $\newline$ The power rule of logarithms states that $n\log_b(a) = \log_b(a^n)$ . Therefore, we can rewrite the expression as $\log_{5}(413^4)$ .
Calculating $413^4$ : Calculate $413^4$ . $\newline$ Using a calculator, we find that $413^4 = 290,541,209$ .
Evaluating the logarithm $\log_{5}(290,541,209)$ : Evaluate the logarithm $\log_{5}(290,541,209)$ . $\newline$ Using a calculator with the change of base formula, we can evaluate the logarithm as $\frac{\log(290,541,209)}{\log(5)}$ .
Performing the calculation: Perform the calculation. $\newline$ Using a calculator, we find that $\log(290,541,209) / \log(5) \approx 14.287$ .
Rounding the answer to the nearest thousandth: Round the answer to the nearest thousandth. $\newline$ The value $14.287$ rounded to the nearest thousandth is $14.287$ .

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