Answer :
To solve the problem of finding [tex]\( x \)[/tex] in the equation [tex]\(\log_5 51.7 = x\)[/tex], we need to use the change of base formula. This formula allows us to calculate logarithms with different bases by converting them into a ratio of logarithms with a common base, often 10.
Here’s a step-by-step guide to solve this:
1. Understand the Change of Base Formula: The change of base formula is:
[tex]\[
\log_b(a) = \frac{\log_c(a)}{\log_c(b)}
\][/tex]
where [tex]\( b \)[/tex] is the base, [tex]\( a \)[/tex] is the number you're taking the logarithm of, and [tex]\( c \)[/tex] is the new base chosen for the calculation (commonly base 10).
2. Define the Problem in Change of Base Terms: For this question, we have:
- [tex]\( a = 51.7 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- We will use base [tex]\( c = 10 \)[/tex].
3. Calculate [tex]\(\log_{10}(51.7)\)[/tex]: This is the logarithm of 51.7 to the base 10. The result is approximately:
[tex]\[
\log_{10}(51.7) \approx 1.7135
\][/tex]
4. Calculate [tex]\(\log_{10}(5)\)[/tex]: This is the logarithm of 5 to the base 10. The result is approximately:
[tex]\[
\log_{10}(5) \approx 0.6990
\][/tex]
5. Apply the Change of Base Formula: Plug the values into the formula:
[tex]\[
\log_5(51.7) = \frac{\log_{10}(51.7)}{\log_{10}(5)}
\][/tex]
[tex]\[
\log_5(51.7) \approx \frac{1.7135}{0.6990}
\][/tex]
6. Calculate the Final Result: Perform the division to find [tex]\( x \)[/tex]:
[tex]\[
x \approx 2.4515
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] in the equation [tex]\(\log_5 51.7 = x\)[/tex] is approximately [tex]\( 2.4515 \)[/tex].
Here’s a step-by-step guide to solve this:
1. Understand the Change of Base Formula: The change of base formula is:
[tex]\[
\log_b(a) = \frac{\log_c(a)}{\log_c(b)}
\][/tex]
where [tex]\( b \)[/tex] is the base, [tex]\( a \)[/tex] is the number you're taking the logarithm of, and [tex]\( c \)[/tex] is the new base chosen for the calculation (commonly base 10).
2. Define the Problem in Change of Base Terms: For this question, we have:
- [tex]\( a = 51.7 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- We will use base [tex]\( c = 10 \)[/tex].
3. Calculate [tex]\(\log_{10}(51.7)\)[/tex]: This is the logarithm of 51.7 to the base 10. The result is approximately:
[tex]\[
\log_{10}(51.7) \approx 1.7135
\][/tex]
4. Calculate [tex]\(\log_{10}(5)\)[/tex]: This is the logarithm of 5 to the base 10. The result is approximately:
[tex]\[
\log_{10}(5) \approx 0.6990
\][/tex]
5. Apply the Change of Base Formula: Plug the values into the formula:
[tex]\[
\log_5(51.7) = \frac{\log_{10}(51.7)}{\log_{10}(5)}
\][/tex]
[tex]\[
\log_5(51.7) \approx \frac{1.7135}{0.6990}
\][/tex]
6. Calculate the Final Result: Perform the division to find [tex]\( x \)[/tex]:
[tex]\[
x \approx 2.4515
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] in the equation [tex]\(\log_5 51.7 = x\)[/tex] is approximately [tex]\( 2.4515 \)[/tex].
