Sure, let's solve the logarithmic expression [tex]\( \log_{125} 5 \)[/tex] step-by-step.
1. Rewrite the Problem Using the Definition of Logarithms:
Recall that [tex]\( \log_b a = c \)[/tex] means that [tex]\( b^c = a \)[/tex]. So here, we have:
[tex]\[
125^c = 5
\][/tex]
2. Express Both Numbers as Powers of a Common Base:
Both 125 and 5 can be expressed as powers of 5:
[tex]\[
125 = 5^3
\][/tex]
So we rewrite the equation [tex]\( 125^c = 5 \)[/tex] with this new representation:
[tex]\[
(5^3)^c = 5
\][/tex]
3. Simplify the Exponential Equation:
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[
5^{3c} = 5^1
\][/tex]
4. Equate the Exponents:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
3c = 1
\][/tex]
5. Solve for [tex]\(c\)[/tex]:
Divide both sides of the equation by 3:
[tex]\[
c = \frac{1}{3}
\][/tex]
Thus, the value of [tex]\( \log_{125} 5 \)[/tex] is [tex]\( \frac{1}{3} \)[/tex].
Converting [tex]\( \frac{1}{3} \)[/tex] to a decimal, we get:
[tex]\[
\log_{125} 5 \approx 0.3333333333333333
\][/tex]
So, the result is:
[tex]\[
\log_{125} 5 = 0.3333333333333333
\][/tex]
