1 Answer

Will Thomson · Answer 1 · 2021-06-19T03:20:55+0000

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;

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

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

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

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

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