Sure, let's determine which of the given equations has [tex]\( x = 4 \)[/tex] as the solution. There are four equations to consider. Let's solve each one step-by-step.
### Equation 1: [tex]\(\log _4(3x+4)=2\)[/tex]
First, we convert the logarithmic equation to its exponential form.
[tex]\[
4^2 = 3x + 4
\][/tex]
Calculate [tex]\(4^2\)[/tex]:
[tex]\[
16 = 3x + 4
\][/tex]
Subtract 4 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
16 - 4 = 3x
\][/tex]
[tex]\[
12 = 3x
\][/tex]
Divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{12}{3} = 4
\][/tex]
Thus, the solution to the first equation is [tex]\(x = 4\)[/tex].
### Equation 2: [tex]\(\log _3(2x-5)=2\)[/tex]
Convert the logarithmic equation to its exponential form:
[tex]\[
3^2 = 2x - 5
\][/tex]
Calculate [tex]\(3^2\)[/tex]:
[tex]\[
9 = 2x - 5
\][/tex]
Add 5 to both sides:
[tex]\[
9 + 5 = 2x
\][/tex]
[tex]\[
14 = 2x
\][/tex]
Divide both sides by 2:
[tex]\[
x = \frac{14}{2} = 7
\][/tex]
The solution to the second equation is [tex]\(x = 7\)[/tex], not 4.
### Equation 3: [tex]\(\log _x 64=4\)[/tex]
Convert the logarithmic equation to its exponential form:
[tex]\[
x^4 = 64
\][/tex]
Take the fourth root of both sides:
[tex]\[
x = \sqrt[4]{64}
\][/tex]
Calculate [tex]\(\sqrt[4]{64}:
\[
x = 2^{\frac{6}{4}} = 2^{1.5} \neq 4
\]
The solution to the third equation is not \(x = 4\)[/tex].
### Equation 4: [tex]\(\log _x 16=4\)[/tex]
Convert the logarithmic equation to its exponential form:
[tex]\[
x^4 = 16
\][/tex]
Take the fourth root of both sides:
[tex]\[
x = \sqrt[4]{16}
\][/tex]
Calculate [tex]\(\sqrt[4]{16}\)[/tex]:
[tex]\[
x = 2
\][/tex]
The solution to the fourth equation is [tex]\(x = 2\)[/tex], not 4.
### Conclusion
Among the four equations, only the first equation [tex]\(\log _4(3 x+4)=2\)[/tex] has [tex]\(x = 4\)[/tex] as the solution. Thus, the correct answer is the first equation.
