To solve the equation [tex]\( e^{x-5} - 2 = 79 \)[/tex], we need to follow these steps:
1. Isolate the exponential term:
[tex]\[
e^{x-5} - 2 = 79
\][/tex]
Add 2 to both sides of the equation:
[tex]\[
e^{x-5} = 81
\][/tex]
2. Take the natural logarithm (ln) of both sides:
Since the natural logarithm function is the inverse of the exponential function, we apply ln to both sides of the equation:
[tex]\[
\ln(e^{x-5}) = \ln(81)
\][/tex]
3. Simplify using properties of logarithms:
The natural logarithm of an exponential function simplifies as follows:
[tex]\[
x - 5 = \ln(81)
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Isolate [tex]\( x \)[/tex] by adding 5 to both sides of the equation:
[tex]\[
x = \ln(81) + 5
\][/tex]
5. Calculate the value of [tex]\( \ln(81) \)[/tex]:
Using a calculator, find the natural logarithm of 81. Let's denote this value as [tex]\( a \)[/tex]:
[tex]\[
a = \ln(81)
\][/tex]
Now, add 5 to [tex]\( a \)[/tex]:
[tex]\[
x = a + 5
\][/tex]
6. Round the final result to the nearest hundredth:
The calculated result is approximately:
[tex]\[
x \approx 9.39
\][/tex]
So, the solution to the equation [tex]\( e^{x-5} - 2 = 79 \)[/tex], rounded to the nearest hundredth, is [tex]\( x \approx 9.39 \)[/tex].
