Answer :

GinnyAnswer
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]

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