Answer :

To solve the equation [tex]\(e^{-x+5}-7=16\)[/tex], let's go through the steps one by one:

1. Start with the original equation:
[tex]\[
e^{-x+5} - 7 = 16
\][/tex]

2. Isolate the exponential expression:
To do this, we'll add 7 to both sides of the equation:
[tex]\[
e^{-x+5} = 23
\][/tex]

3. Solve for [tex]\(-x + 5\)[/tex]:
To solve for the exponent, we'll take the natural logarithm (ln) of both sides. This helps because the natural logarithm and the exponential function are inverse operations:
[tex]\[
-x + 5 = \ln(23)
\][/tex]

4. Solve for [tex]\(x\)[/tex]:
Rearrange the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
-x = \ln(23) - 5
\][/tex]
Multiply by -1 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 5 - \ln(23)
\][/tex]

Thus, the solution to the equation is [tex]\(x = 5 - \ln(23)\)[/tex].

5. Verify by substituting [tex]\(x = 1\)[/tex]:
Let's evaluate the expression at [tex]\(x = 1\)[/tex] to verify:
[tex]\[
e^{-1 + 5} - 7 = e^{4} - 7
\][/tex]
This shows how the expression behaves when [tex]\(x = 1\)[/tex].

So, the solution to the equation is [tex]\(x = 5 - \ln(23)\)[/tex], and at [tex]\(x = 1\)[/tex], the expression evaluates to [tex]\(e^4 - 7\)[/tex].

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