Question

The value of \( 3^{3-\log_3 5} \) is

• A. \( \frac{5}{9} \)
• B. \( \frac{27}{5} \)
• C. \( \frac{9}{5} \)
• D. \( \frac{5}{27} \)

Hint:

The two key logarithm rules for this problem are the exponent rule for subtraction, \( a^{m-n} = a^m/a^n \), and the fundamental inverse property, \( a^{\log_a x} = x \). Applying these in order simplifies the expression directly.

Answer 1

The Correct Option is B

We need to evaluate the expression \( 3^{3-\log_3 5} \). Using the exponent rule \( a^{m-n} = \frac{a^m}{a^n} \), we can split the expression: \[ 3^{3-\log_3 5} = \frac{3^3}{3^{\log_3 5}} \] Now, we evaluate the numerator and the denominator separately. Numerator: \( 3^3 = 27 \). Denominator: We use the fundamental logarithm identity \( a^{\log_a x} = x \). \[ 3^{\log_3 5} = 5 \] Now substitute these values back into the fraction: \[ \frac{27}{5} \]