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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.
[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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