Question

Evaluate the value of the following expression:
\[ 3^{\left(2+\log_3 5\right)} \div \log_{25} 125 \]

• A. 75
• B. 40
• C. 15
• D. 60

Hint:

Always convert logarithmic bases into the same base to simplify calculations easily.

Answer 1

The Correct Option is A

Step 1: Simplify the exponent in the numerator.
\[ 3^{\left(2+\log_3 5\right)} = 3^2 \cdot 3^{\log_3 5} \]
Using the identity \(a^{\log_a b} = b\), we get:
\[ 3^2 \cdot 5 = 9 \cdot 5 = 45 \] Step 2: Simplify the denominator.
\[ \log_{25} 125 \]
Write both numbers in terms of base \(5\):
\[ 25 = 5^2,\quad 125 = 5^3 \]
\[ \log_{25} 125 = \frac{\log 5^3}{\log 5^2} = \frac{3}{2} \] Step 3: Divide the results.
\[ \frac{45}{\frac{3}{2}} = 45 \times \frac{2}{3} = 30 \] Final Answer:
\[ \boxed{30} \]