Question20 Mark for Review 425
The function $f$ is given by $f(x)=\log _{4}\left(\frac{16}{x}\right)$. Of the following, solving which equation gives the solution to $f(x)=5$ ? (A) $\log _{4} 16-\ln \left(\frac{x}{4}\right)=5$ (B) $\log _{4} 16-\frac{\ln x}{\ln 4}=5$ (C) $\ln \left(\frac{x}{4}\right)-\log _{4} 16=5$ (D) $\frac{\ln x}{\ln 4}-\log _{4} 16=5$

Studdy Solution
Compare the simplified equation to the given options.
First, solve $\frac{16}{x} = 4^5$.
$$4^5 = 1024$$
So, $\frac{16}{x} = 1024$.
Rearrange to find $x$:
$$x = \frac{16}{1024}$$
Now, let's compare the given options:
(A) $\log _{4} 16-\ln \left(\frac{x}{4}\right)=5$
(B) $\log _{4} 16-\frac{\ln x}{\ln 4}=5$
(C) $\ln \left(\frac{x}{4}\right)-\log _{4} 16=5$
(D) $\frac{\ln x}{\ln 4}-\log _{4} 16=5$
The correct equation should reflect the transformation of the original equation $\log_{4}\left(\frac{16}{x}\right) = 5$.
Using the property $\log_{b}(a) = \frac{\ln a}{\ln b}$, we can rewrite:
$$\log_4\left(\frac{16}{x}\right) = \frac{\ln \left(\frac{16}{x}\right)}{\ln 4}$$
This implies:
$$\frac{\ln 16 - \ln x}{\ln 4} = 5$$
Rearrange to match one of the options:
$$\frac{\ln x}{\ln 4} = \ln 16 - 5$$
This matches option (B):
$$\log _{4} 16-\frac{\ln x}{\ln 4}=5$$
The solution is option (B).