Find the logarithm using common logarithms and the change-of-base formula.

[tex]\[
\begin{array}{c}
\log_{500} 50 \\
\log_{500} 50 =
\end{array}
\][/tex]

(Simplify your answer. Round to four decimal places as needed.)



Answer :

To find [tex]\(\log_{500} 50\)[/tex] using common logarithms and the change-of-base formula, follow these steps:

1. Understand the Change-of-Base Formula: The change-of-base formula for logarithms states that:
[tex]\[ \log_b a = \frac{\log_c a}{\log_c b} \][/tex]
Where [tex]\( b \)[/tex] is the base of the logarithm you want to change, [tex]\( a \)[/tex] is the value you are taking the logarithm of, and [tex]\( c \)[/tex] is the new base (commonly base 10 or base [tex]\( e \)[/tex]).

2. Apply the Change-of-Base Formula:
Choose base [tex]\( c = 10 \)[/tex] (common logarithm). Then,
[tex]\[ \log_{500} 50 = \frac{\log_{10} 50}{\log_{10} 500} \][/tex]

3. Calculate the Common Logarithms:
Determine the values of [tex]\(\log_{10} 50\)[/tex] and [tex]\(\log_{10} 500\)[/tex]:
[tex]\[ \log_{10} 50 = 1.69897 \quad \text{(approximately)} \][/tex]
[tex]\[ \log_{10} 500 = 2.69897 \quad \text{(approximately)} \][/tex]

4. Divide the Logarithms:
Use these values in the change-of-base formula:
[tex]\[ \log_{500} 50 = \frac{\log_{10} 50}{\log_{10} 500} = \frac{1.69897}{2.69897} \][/tex]

5. Perform the Division:
Carry out the division to obtain the result:
[tex]\[ \frac{1.69897}{2.69897} \approx 0.6294882868674145 \][/tex]

6. Round the Result:
Round the result to four decimal places:
[tex]\[ \log_{500} 50 \approx 0.6295 \][/tex]

So, [tex]\(\log_{500} 50\)[/tex] rounded to four decimal places is:
[tex]\[ \log_{500} 50 \approx 0.6295 \][/tex]