your equation is, as far as i can tell,
3e^(-x) + 11 = 48
if so, subtract 11 from both sides of the equation to get:
3e^(-x) = 37
divide both sides of the equation by 3 to get:
e^(-x) = 37/3
take thre natural log of both sides of the equation to get:
ln(e^(-x) = ln(37/3)
since ln(e^(-x) = -x * ln(e) and, since ln(e) = 1, then ln(e^(-x) = -x.
your equation becomeds:
-x = ln(37/3)
solve for x to get:
x = -ln(37/3) = -2.512305624.
confirm by replacing x in the original equation to get:
3e^(-x) + 11 = 48 becomes 3e^(-2.512305624) + 11 = 48
evaluate to get 48 = 48, confirming the value of x is correct.
HOWEVER,
it is also possible that your equation is:
3e^(−x+11) = 48
without any parentheses, it's difficult to determine what you meant.
in this case, divide both sides of the equation by 3 to get:
e^(-x+11) = 16
take the natural log of both sides of the equation to get:
ln(e^(-x+11)) = ln(16)
since ln(e^(-x+11)) = (-x+11) * ln(e) which is equal to -x+11, your equation becomes:
-x + 11 = ln(16).
subtract 11 from both sides of the equation to get:
-x = ln(16) - 11.
multiply both sides of the equation by -1 to get:
x = -ln(16) + 11
solve for x to get:
x = 8.227411278.
confirm by replacing x in the original equation to get:
3e^(−x+11) = 48 becomes 3e^(−8.227411278+11) = 48
evaluate to get 48 = 48, confirming that the value of x is good.
if the original equation was 3e^(-x) + 11 = 48, then the solution is x = -2.512305624.
if the original equation was 3e^(−x+11) = 48, then the solution is x = 8.227411278.
i would have chosen the first option of 3e^−x+11 = 48 = 3e^(-x) + 11 = 48 because, without the parentheses, the equation becomes:
3 * e^(-x) + 11 = 48.
it then becomes:
(3 * e^(-x) + 11 = 48
subtract 11 from both sides to get:
3 * e^(-x) = 37
divide both sides by 3 to get:
e^(-x) = 37/3.
you then took the natural log of both sides to get:
-x = ln(37/3).
you then solve for x to get:
x = -ln(37/3).
these equations can be graphed.
the graph is shown below:
the red equation is y = 3 * e^(-x + 11).
the blue equation is y = 3 * e^(-x) + 11.
