To determine which of the given logarithmic equations have [tex]\( x = 2 \)[/tex] as the solution, let's analyze each equation step-by-step.
### Equation 1: [tex]\(\log_2(5x + 6) = 4\)[/tex]
1. Substitute [tex]\( x = 2 \)[/tex] into the equation:
[tex]\[
\log_2(5(2) + 6) = 4
\][/tex]
2. Simplify inside the logarithm:
[tex]\[
\log_2(10 + 6) = 4 \implies \log_2(16) = 4
\][/tex]
3. Verify if this is true:
[tex]\[
2^4 = 16
\][/tex]
This is correct.
### Equation 2: [tex]\(\log_x 16 = 4\)[/tex]
1. Substitute [tex]\( x = 2 \)[/tex] into the equation:
[tex]\[
\log_2 16 = 4
\][/tex]
2. Verify if this is true:
[tex]\[
2^4 = 16
\][/tex]
This is correct.
### Equation 3: [tex]\(\log_3(6x + 4) = 3\)[/tex]
1. Substitute [tex]\( x = 2 \)[/tex] into the equation:
[tex]\[
\log_3(6(2) + 4) = 3
\][/tex]
2. Simplify inside the logarithm:
[tex]\[
\log_3(12 + 4) = 3 \implies \log_3(16) = 3
\][/tex]
3. Verify if this is true:
[tex]\[
3^3 = 27 \quad (\text{but } 3^3 \neq 16)
\][/tex]
This is not correct.
### Equation 4: [tex]\(\log_x 36 = 6\)[/tex]
1. Substitute [tex]\( x = 2 \)[/tex] into the equation:
[tex]\[
\log_2 36 = 6
\][/tex]
2. Verify if this is true:
[tex]\[
2^6 = 64 \quad (\text{but } 2^6 \neq 36)
\][/tex]
This is not correct.
### Conclusion
Based on our analysis, the solution [tex]\( x = 2 \)[/tex] satisfies the following equations:
1. [tex]\(\log_2(5x + 6) = 4\)[/tex]
2. [tex]\(\log_x 16 = 4\)[/tex]
The equations that do not have [tex]\( x = 2 \)[/tex] as a solution are:
1. [tex]\(\log_3(6x + 4) = 3\)[/tex]
2. [tex]\(\log_x 36 = 6\)[/tex]
