Answer :
To solve the equation [tex]\( e^{-x+5} - 7 = 16 \)[/tex], follow these steps:
1. Isolate the exponential expression:
Begin by simplifying the equation to isolate the exponential term on one side. Start with the original equation:
[tex]\[
e^{-x+5} - 7 = 16
\][/tex]
Add 7 to both sides to move the constant term over:
[tex]\[
e^{-x+5} = 23
\][/tex]
2. Apply the natural logarithm:
To solve for [tex]\( x \)[/tex], take the natural logarithm (ln) of both sides. The natural logarithm is the inverse operation of the exponential function.
[tex]\[
\ln(e^{-x+5}) = \ln(23)
\][/tex]
Using the property of logarithms that [tex]\( \ln(e^a) = a \)[/tex], we find:
[tex]\[
-x+5 = \ln(23)
\][/tex]
3. 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]
Finally, multiply both sides by -1 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 5 - \ln(23)
\][/tex]
So, the solution to the equation is [tex]\( x = 5 - \log(23) \)[/tex]. This expression gives the value of [tex]\( x \)[/tex] that satisfies the original equation.
1. Isolate the exponential expression:
Begin by simplifying the equation to isolate the exponential term on one side. Start with the original equation:
[tex]\[
e^{-x+5} - 7 = 16
\][/tex]
Add 7 to both sides to move the constant term over:
[tex]\[
e^{-x+5} = 23
\][/tex]
2. Apply the natural logarithm:
To solve for [tex]\( x \)[/tex], take the natural logarithm (ln) of both sides. The natural logarithm is the inverse operation of the exponential function.
[tex]\[
\ln(e^{-x+5}) = \ln(23)
\][/tex]
Using the property of logarithms that [tex]\( \ln(e^a) = a \)[/tex], we find:
[tex]\[
-x+5 = \ln(23)
\][/tex]
3. 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]
Finally, multiply both sides by -1 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 5 - \ln(23)
\][/tex]
So, the solution to the equation is [tex]\( x = 5 - \log(23) \)[/tex]. This expression gives the value of [tex]\( x \)[/tex] that satisfies the original equation.
