Question
Solve for x in the logarithmic equation
5
5
(
625
)
=
5
x
5
log
5
(625)
=5
x
.
5
5
(
625
)
=
5
x
5
log
5
(625)
=5
x
.
Answers
GPT-4o mini
To solve the equation \( 5 \log_5(625) = 5^x \), we start by simplifying the left-hand side.
First, we calculate \( \log_5(625) \). We can express 625 as a power of 5:
\[
625 = 5^4
\]
Thus,
\[
\log_5(625) = \log_5(5^4) = 4
\]
Now, substituting this value back into the equation gives:
\[
5 \log_5(625) = 5 \cdot 4 = 20
\]
So the equation becomes:
\[
20 = 5^x
\]
To solve for \( x \), we can express 20 as a power of 5. However, since 20 is not a power of 5, we take the logarithm base 5 on both sides:
\[
\log_5(20) = \log_5(5^x)
\]
Using the properties of logarithms, we simplify the right side:
\[
\log_5(20) = x
\]
Thus, we have:
\[
x = \log_5(20)
\]
We can convert this to a base-10 logarithm using the change of base formula if needed:
\[
x = \frac{\log_{10}(20)}{\log_{10}(5)}
\]
However, if you just need the answer, we can leave it as:
\[
\boxed{\log_5(20)}
\]
First, we calculate \( \log_5(625) \). We can express 625 as a power of 5:
\[
625 = 5^4
\]
Thus,
\[
\log_5(625) = \log_5(5^4) = 4
\]
Now, substituting this value back into the equation gives:
\[
5 \log_5(625) = 5 \cdot 4 = 20
\]
So the equation becomes:
\[
20 = 5^x
\]
To solve for \( x \), we can express 20 as a power of 5. However, since 20 is not a power of 5, we take the logarithm base 5 on both sides:
\[
\log_5(20) = \log_5(5^x)
\]
Using the properties of logarithms, we simplify the right side:
\[
\log_5(20) = x
\]
Thus, we have:
\[
x = \log_5(20)
\]
We can convert this to a base-10 logarithm using the change of base formula if needed:
\[
x = \frac{\log_{10}(20)}{\log_{10}(5)}
\]
However, if you just need the answer, we can leave it as:
\[
\boxed{\log_5(20)}
\]