Option B: [tex]\ln 6=5 x[/tex] is the correct answer.
Explanation:
The exponential equation is [tex]e^{5 x}=6[/tex]
If [tex]f(x)=g(x)[/tex], then [tex]\ln (f(x))=\ln (g(x))[/tex]
Thus, the equation becomes
[tex]\ln \left(e^{5 x}\right)=\ln (6)[/tex]
Applying log rule, [tex]\log _{a}\left(x^{b}\right)=b \cdot \log _{a}(x)[/tex] and thus the equation becomes
[tex]5 x \ln (e)=\ln (6)[/tex]
Since, we know that, [tex]\ln (e)=1[/tex], using this we get,
[tex]5 x=\ln (6)[/tex]
Hence, the logarithmic equation which is equivalent to the exponential equation [tex]e^{5 x}=6[/tex] is [tex]\ln 6=5 x[/tex]
Thus, Option B is the correct answer.
