Question

Find the value of \( (125)^{\log_{625} 5} \)

• A. \(15\)
• B. \(25\)
• C. \(5\sqrt{5}\)
• D. \(3\)
• E. None

Hint:

When logarithms and exponents appear together, first express all numbers using the same base. This often simplifies the logarithmic expression significantly.

Answer 1

Answer: (E)

Concept: To evaluate expressions involving logarithms and exponents, rewrite the numbers in terms of the same base. \[ 125 = 5^3, \qquad 625 = 5^4 \] Also, using the property: \[ \log_{a^m}(b^n) = \frac{n}{m}\log_a b \]

Step 1: Rewrite the given expression using base \(5\). \[ (125)^{\log_{625}5} = (5^3)^{\log_{5^4}5} \]

Step 2: Evaluate the logarithm. \[ \log_{5^4}5 = \frac{1}{4}\log_5 5 = \frac{1}{4} \]

Step 3: Substitute the value. \[ (5^3)^{1/4} = 5^{3/4} \]

Step 4: Evaluate the final expression. The value \(5^{3/4} = \sqrt[4]{125}\) is approximately \(3.34\). Since this value does not match any of the options given, the correct choice is \[ \text{None} \]