Evaluate the logarithm. Round your answer to the nearest thousandth. log_(4)(0.6)~~

Problem Understanding: Understand the problem. We need to find the value of the logarithm of 0.6 with base 4, which is written as log₄(0.6). This means we are looking for the power to which 4 must be raised to get 0.6.Change of Base Formula: Use the change of base formula. The change of base formula allows us to write log₄(0.6) in terms of a logarithm with a more common base, such as base 10 or base e (natural logarithm). We will use base 10 for this calculation. The change of base formula is: logₐ(b) = log_c(b) / log_c(a) So, log₄(0.6) = log₁₀(0.6) / log₁₀(4)Calculating Values: Calculate the values using a calculator. Using a calculator, we find: log₁₀(0.6) ≈ -0.2218 (to four decimal places) log₁₀(4) ≈ 0.6021 (to four decimal places) Now we divide these two values to find log₄(0.6). log₄(0.6) ≈ -0.2218 / 0.6021Performing Division: Perform the division. -0.2218 / 0.6021 ≈ -0.368 (rounded to the nearest thousandth)Result Checking: Check the result. Since 4 raised to any positive power will be greater than 1, and 4 raised to any negative power will be less than 1, it makes sense that log₄(0.6) is negative because 0.6 is less than 1. This is a sanity check for our result.

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

log_(4)(0.6)~~

Evaluate the logarithm. $\newline$ Round your answer to the nearest thousandth. $\newline$ $\log _{4}(0.6) \approx$

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Algebra 1

Compare linear, exponential, and quadratic growth

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Q. Evaluate the logarithm.

\newline

Round your answer to the nearest thousandth.

\newline

\log _{4}(0.6) \approx

Problem Understanding: Understand the problem. $\newline$ We need to find the value of the logarithm of $0.6$ with base $4$ , which is written as $\log_4(0.6)$ . This means we are looking for the power to which $4$ must be raised to get $0.6$ .
Change of Base Formula: Use the change of base formula. $\newline$ The change of base formula allows us to write $\log_4(0.6)$ in terms of a logarithm with a more common base, such as base $10$ or base $e$ (natural logarithm). We will use base $10$ for this calculation. $\newline$ The change of base formula is: $\newline$ $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$ $\newline$ So, $\log_4(0.6) = \frac{\log_{10}(0.6)}{\log_{10}(4)}$
Calculating Values: Calculate the values using a calculator. $\newline$ Using a calculator, we find: $\newline$ $\log_{10}(0.6) \approx -0.2218$ (to four decimal places) $\newline$ $\log_{10}(4) \approx 0.6021$ (to four decimal places) $\newline$ Now we divide these two values to find $\log_{4}(0.6)$ . $\newline$ $\log_{4}(0.6) \approx \frac{-0.2218}{0.6021}$
Performing Division: Perform the division. $\newline$ $-0.2218 / 0.6021 \approx -0.368$ (rounded to the nearest thousandth)
Result Checking: Check the result. $\newline$ Since $4$ raised to any positive power will be greater than $1$ , and $4$ raised to any negative power will be less than $1$ , it makes sense that $\log_4(0.6)$ is negative because $0.6$ is less than $1$ . This is a sanity check for our result.