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Math  /  Algebra

Question20 Mark for Review 425
The function ff is given by f(x)=log⁡4(16x)f(x)=\log _{4}\left(\frac{16}{x}\right). Of the following, solving which equation gives the solution to f(x)=5f(x)=5 ? (A) log⁡416−ln⁡(x4)=5\log _{4} 16-\ln \left(\frac{x}{4}\right)=5 (B) log⁡416−ln⁡xln⁡4=5\log _{4} 16-\frac{\ln x}{\ln 4}=5 (C) ln⁡(x4)−log⁡416=5\ln \left(\frac{x}{4}\right)-\log _{4} 16=5 (D) ln⁡xln⁡4−log⁡416=5\frac{\ln x}{\ln 4}-\log _{4} 16=5

Studdy Solution

STEP 1

1. The function f(x)=log⁡4(16x) f(x) = \log_{4}\left(\frac{16}{x}\right) .
2. We need to solve for x x when f(x)=5 f(x) = 5 .
3. We are given four options and need to determine which equation correctly represents the solution to f(x)=5 f(x) = 5 .

STEP 2

1. Set up the equation f(x)=5 f(x) = 5 .
2. Use properties of logarithms to simplify the equation.
3. Compare the simplified equation to the given options.

STEP 3

Set up the equation f(x)=5 f(x) = 5 .
Given f(x)=log⁡4(16x) f(x) = \log_{4}\left(\frac{16}{x}\right) , we set:
log⁡4(16x)=5 \log_{4}\left(\frac{16}{x}\right) = 5

STEP 4

Use properties of logarithms to simplify the equation.
Using the property log⁡b(a)=c \log_{b}(a) = c implies a=bc a = b^c , we have:
16x=45 \frac{16}{x} = 4^5

STEP 5

Compare the simplified equation to the given options.
First, solve 16x=45 \frac{16}{x} = 4^5 .
45=1024 4^5 = 1024
So, 16x=1024 \frac{16}{x} = 1024 .
Rearrange to find x x :
x=161024 x = \frac{16}{1024}
Now, let's compare the given options:
(A) log⁡416−ln⁡(x4)=5\log _{4} 16-\ln \left(\frac{x}{4}\right)=5
(B) log⁡416−ln⁡xln⁡4=5\log _{4} 16-\frac{\ln x}{\ln 4}=5
(C) ln⁡(x4)−log⁡416=5\ln \left(\frac{x}{4}\right)-\log _{4} 16=5
(D) ln⁡xln⁡4−log⁡416=5\frac{\ln x}{\ln 4}-\log _{4} 16=5
The correct equation should reflect the transformation of the original equation log⁡4(16x)=5 \log_{4}\left(\frac{16}{x}\right) = 5 .
Using the property log⁡b(a)=ln⁡aln⁡b\log_{b}(a) = \frac{\ln a}{\ln b}, we can rewrite:
log⁡4(16x)=ln⁡(16x)ln⁡4 \log_4\left(\frac{16}{x}\right) = \frac{\ln \left(\frac{16}{x}\right)}{\ln 4}
This implies:
ln⁡16−ln⁡xln⁡4=5 \frac{\ln 16 - \ln x}{\ln 4} = 5
Rearrange to match one of the options:
ln⁡xln⁡4=ln⁡16−5 \frac{\ln x}{\ln 4} = \ln 16 - 5
This matches option (B):
log⁡416−ln⁡xln⁡4=5 \log _{4} 16-\frac{\ln x}{\ln 4}=5
The solution is option (B).

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