Question

Solve for x write the exact answer using either base -10 or base-e logarithms

Answer 1

Given:

The given expression is $2^{x+5}=13^{2 x}$

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 $f(x)=g(x)$ then $\ln (f(x))=\ln (g(x))$

Thus, we get;

$\ln \left(2^{x+5}\right)=\ln \left(13^{2 x}\right)$

Applying the log rule, $\log _{a}\left(x^{b}\right)=b \cdot \log _{a}(x)$, we get;

$(x+5) \ln (2)=2 x \ln (13)$

Expanding, we get;

$x \ln (2)+5 \ln (2)=2 x \ln (13)$

Subtracting both sides by $5 \ln (2)$, we get;

$x \ln (2)=2 x \ln (13)-5 \ln (2)$

Subtracting both sides by $2 x \ln (13)$, we get;

$x \ln (2)-2 x \ln (13)=-5 \ln (2)$

Taking out the common term x, we have;

$x( \ln (2)-2 \ln (13))=-5 \ln (2)$

                          $x=\frac{-5 \ln (2)}{\ln (2)-2 \ln (13)}$

                          $x=\frac{5 \ln (2)}{2 \ln (13)-\ln (2)}$

Thus, the value of x is $x=\frac{5 \ln (2)}{2 \ln (13)-\ln (2)}$