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Answer :
To solve the equation [tex]\( e^{-x+5} - 7 = 16 \)[/tex], let's go through the problem step by step.
1. Isolate the Exponential Term:
The equation is [tex]\( e^{-x+5} - 7 = 16 \)[/tex]. Start by adding 7 to both sides to isolate the exponential component:
[tex]\[
e^{-x+5} = 23
\][/tex]
2. Take the Natural Logarithm:
To solve for [tex]\(-x + 5\)[/tex], take the natural logarithm on both sides:
[tex]\[
\ln(e^{-x+5}) = \ln(23)
\][/tex]
3. Simplify Using Logarithm Properties:
The natural logarithm and the exponential function are inverse operations, so the left-hand side simplifies to:
[tex]\[
-x + 5 = \ln(23)
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Rearrange the equation to solve for [tex]\( x \)[/tex]. Start by subtracting 5 from both sides:
[tex]\[
-x = \ln(23) - 5
\][/tex]
Then, multiply by -1 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 5 - \ln(23)
\][/tex]
Therefore, the solution to the equation is [tex]\( x = 5 - \ln(23) \)[/tex].
1. Isolate the Exponential Term:
The equation is [tex]\( e^{-x+5} - 7 = 16 \)[/tex]. Start by adding 7 to both sides to isolate the exponential component:
[tex]\[
e^{-x+5} = 23
\][/tex]
2. Take the Natural Logarithm:
To solve for [tex]\(-x + 5\)[/tex], take the natural logarithm on both sides:
[tex]\[
\ln(e^{-x+5}) = \ln(23)
\][/tex]
3. Simplify Using Logarithm Properties:
The natural logarithm and the exponential function are inverse operations, so the left-hand side simplifies to:
[tex]\[
-x + 5 = \ln(23)
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Rearrange the equation to solve for [tex]\( x \)[/tex]. Start by subtracting 5 from both sides:
[tex]\[
-x = \ln(23) - 5
\][/tex]
Then, multiply by -1 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 5 - \ln(23)
\][/tex]
Therefore, the solution to the equation is [tex]\( x = 5 - \ln(23) \)[/tex].
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