math inverse

Question

(2e^x-3)/(7e^x+9)

Answer 1

It's not clear how the problem reads.

If it's (2e^(x-3))/(7e^(x+9)) then it = 2/e^12 = about 1.65 x 10^-6

but if it reads (2e^x -3)/(7e^x +9) then see the other answers

Answer 2

Hi steven,

Probably this will help you

Let y =2e x-3/7e^x+9

Multiplying both sides by 7e^x+9

(7e^x+9) y = 2e^x-3

Apply distributive property on left side of equation

7e^xy+9y = 2e^x-3

Subtracting 7e^xy on both sides

9y =2e^x-3-7e^xy

Now adding 3 to both sides

9y+3=2e^x-7e^xy

Taking e^x as common factor on right side

9y+3=e^x(2-7y)

Dividing both sides by 2-7y

(9y+3)/ (2-7y) =e^x

Taking natural logarithm (ln) on both sides

ln[(9y+3)/(2-7y)]= lne^x =x

hence,

x = ln(9y+3)-ln(2-7y)                   

now using  ln(a/b)= ln(a)-ln(b)

above can also be written as

ln(9x+3)-ln(2-7x) [changing of variable does not affect the function]

which is the requied inverse.

Answer 3

Steven, first of all, exchange for all x's for y's.

y = (2e^x- 3)/(7e^x + 9)    =>   x = (2e^y - 3)/(7e^y+ 9)

The next step is to multiply both sides for 7e^y + 9; it's supposed to give you

x(7e^y+ 9) = 2e^y - 3

Apply now the distributive property for the left side of the equation.

7xe^y + 9x = 2e^y - 3

Next, subtract 2e^yon both sides.

7xe^y + 9x - 2e^y= 2e^y- 3 - 2e^y

7xe^y + 9x - 2e^y = ^-3

Next, subtract 9x on both sides.

7xe^y + 9x - 2e^y - 9x = ^-3 - 9x

7xe^y - 2e^y = ^-9x - 3

Having the terms with e^yon one side of the equation, factor the left side using, precisely, e^y as the common factor.

e^y(7x - 2) = ^-9x - 3

Now divide each side by 7x - 2; the equation now must be solved for e^y.

e^y = (^-9x - 3)/(7x - 2)

It is preferable to factor the numerator using ^-1 as the common factor. Note: Brackets instead of parentheses where needed in order to distinguish the numerator from the denominator.

e^y = [^-1(3 - 9x)]/(7x - 2)

Now, apply the ^-1 to the denominator.

e^y = (3 - 9x)/(2 - 7x)

For the final step, you must get rid of the e in e^y. For this to happen, you must use the natural logarithm (ln), which is some sort of an opposite operation for e. Note: The brackets are also used for this step, this time to distinguish the content of the natural logarithm.

y = ln[(3 - 9x)/(2 - 7x)]

Seeing that this equation is finally solved for y, you might say that this is the right answer, but, to make this result a lot more classy, let's use the property of logarithms concerning division, that is, log(a/b) = log(a) - log(b). Applying this law to the result, this gives you

y = ln(3 - 9x) - ln(2 - 7x)

This result is the inverse of the original exercise.