Question

select the correct answer.
which equation can be solved using the inverse property?
a. \\(log_{2} x = log_{2} 6\\)
b. \\(e^{3.5} = e^{2x}\\)
c. \\(10^{2x} = 10^{-8}\\)
d. \\(log_{4} x = -1\\)

Explanation:

Step1: Recall Inverse Property

The inverse property for logarithms and exponents: If $\log_b a = c$, then $b^c = a$ (inverse of log), and if $b^x = b^y$, then $x = y$ (for same base, not inverse property). The inverse property here refers to using the inverse function (exponential for log, log for exponential) to solve.

Step2: Analyze Option A

$\log_2 x = \log_2 6$: Since the logs have the same base and are equal, we can conclude $x = 6$ by the one - to - one property of logarithms, not the inverse property.

Step3: Analyze Option B

$e^{3.5}=e^{2x}$: Since the exponentials have the same base and are equal, we can conclude $3.5 = 2x$ by the one - to - one property of exponential functions, not the inverse property.

Step4: Analyze Option C

$10^{2x}=10^{- 8}$: Since the exponentials have the same base and are equal, we can conclude $2x=-8$ by the one - to - one property of exponential functions, not the inverse property.

Step5: Analyze Option D

$\log_4 x=-1$: To solve for $x$, we use the inverse property of logarithms. The inverse of $\log_b x$ is $b^y=x$ (where $y = \log_b x$). So, if $\log_4 x=-1$, then by the definition of a logarithm (inverse property of log and exponential), we can rewrite it as $x = 4^{-1}=\frac{1}{4}$. This uses the inverse property (converting log to exponential, which is the inverse operation).

Answer:

D. $\log_4 x = - 1$