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STatiana [176]
3 years ago
8

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

Mathematics
1 answer:
jeka57 [31]3 years ago
6 0

<u>Given</u>:

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.

<u>Value of x:</u>

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)}

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