Answer :
Given:
The given expression is [tex]2^{x+5}=13^{2 x}[/tex]
We need to determine the value of x using either base - 10 or base - e logarithms.
Value of x:
Let us determine the value of x using the base - e logarithms.
Applying the log rule that if [tex]f(x)=g(x)[/tex] then [tex]\ln (f(x))=\ln (g(x))[/tex]
Thus, we get;
[tex]\ln \left(2^{x+5}\right)=\ln \left(13^{2 x}\right)[/tex]
Applying the log rule, [tex]\log _{a}\left(x^{b}\right)=b \cdot \log _{a}(x)[/tex], we get;
[tex](x+5) \ln (2)=2 x \ln (13)[/tex]
Expanding, we get;
[tex]x \ln (2)+5 \ln (2)=2 x \ln (13)[/tex]
Subtracting both sides by [tex]5 \ln (2)[/tex], we get;
[tex]x \ln (2)=2 x \ln (13)-5 \ln (2)[/tex]
Subtracting both sides by [tex]2 x \ln (13)[/tex], we get;
[tex]x \ln (2)-2 x \ln (13)=-5 \ln (2)[/tex]
Taking out the common term x, we have;
[tex]x( \ln (2)-2 \ln (13))=-5 \ln (2)[/tex]
[tex]x=\frac{-5 \ln (2)}{\ln (2)-2 \ln (13)}[/tex]
[tex]x=\frac{5 \ln (2)}{2 \ln (13)-\ln (2)}[/tex]
Thus, the value of x is [tex]x=\frac{5 \ln (2)}{2 \ln (13)-\ln (2)}[/tex]