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Solve the exponential equation using logarithms. Round decimals to 4 places.

[tex]2^{x-5} = 36.6[/tex]

A. 8.6
B. 6.5635
C. 7.2204
D. 10.1938

Answer :

To solve the equation

[tex]$$
2^{x-5} = 36.6,
$$[/tex]

follow these steps:

1. Take the logarithm of both sides:
You can take the natural logarithm (or any logarithm) on both sides of the equation:

[tex]$$
\ln\left(2^{x-5}\right) = \ln(36.6).
$$[/tex]

2. Apply the power rule of logarithms:
The logarithm of a power gives you:

[tex]$$
(x-5)\ln(2) = \ln(36.6).
$$[/tex]

3. Solve for [tex]\(x-5\)[/tex]:
Isolate [tex]\(x-5\)[/tex] by dividing both sides by [tex]\(\ln(2)\)[/tex]:

[tex]$$
x-5 = \frac{\ln(36.6)}{\ln(2)}.
$$[/tex]

A calculator evaluation shows that

[tex]$$
\frac{\ln(36.6)}{\ln(2)} \approx 5.1938.
$$[/tex]

4. Solve for [tex]\(x\)[/tex]:
Add 5 to both sides to get [tex]\(x\)[/tex]:

[tex]$$
x = 5 + \frac{\ln(36.6)}{\ln(2)}.
$$[/tex]

Substituting the numerical value:

[tex]$$
x \approx 5 + 5.1938 = 10.1938.
$$[/tex]

Thus, the solution to the equation is

[tex]$$
x \approx 10.1938.
$$[/tex]

This detailed step-by-step solution shows how logarithms are used to isolate and solve for [tex]\(x\)[/tex] in the exponential equation.

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