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]
