Question

an equation involves the expression \\(\log_{9}(27^{x})\\), which is equivalent to a rational multiple of \\(x\\). by rewriting the expression in an equivalent form, the value of the rational number can be determined without use of a calculator or complicated calculations. which of the following is an equivalent expression that satisfies this requirement?

a \\(x\ln\left(\frac{27}{9}\
ight)\\)

b \\(x\log_{3}\left(\frac{27}{9}\
ight)\\)

c \\(\frac{x\ln 27}{\ln 9}\\)

d \\(\frac{x\log_{3}27}{\log_{3}9}\\)

Explanation:

Step1: Apply logarithm power rule

$\log_9(27^x) = x \log_9 27$

Step2: Apply change of base formula

$\log_9 27 = \frac{\log_3 27}{\log_3 9}$

Step3: Combine expressions

$x \log_9 27 = x \cdot \frac{\log_3 27}{\log_3 9}$

Step4: Verify rational multiple

$\log_3 27 = 3$, $\log_3 9 = 2$, so $\frac{3}{2}$ is rational, satisfying the requirement. Options A and C use natural logs (not base 3, harder to compute rational value), B has incorrect base in the log term. Thus D is correct.

Answer:

D. $\frac{x \log_3 27}{\log_3 9}$