Solve the Equations of the Exponential Functions

[Bin222]

Hello guys, do you have any advise on how to approach the following problems and which rules should be applied?

1. Solve:

a) lnx+ln(x-1)=1

b) 11(2+e^-x)^-1=4

2. Find the inverse of y=(lnx)^3

 

[HallsofIvy]

Hello guys, do you have any advise on how to approach the following problems and which rules should be applied?

1. Solve:

a) lnx+ln(x-1)=1

ln(x)+ ln(x- 1)= ln(x(x- 1))= 1. Get rid of the ln by using its inverse "e^x".

b) 11(2+e^-x)^-1=4

Start as you would with any algebra equation- divide both sides by 11 to get (2+ e^(-x))^{-1}= 4/11. Of course, (2+ e^(-x))^(-1)= 1/(2+ e^(-x))= 4/11 so "invert" both sides:
2+ e^(-x)= 11/4. Can you continue from here? (The inverse of e^x is ln).

2. Find the inverse of y=(lnx)^3

Swap x and y: x= (ln(y))^3 and solve for y. An obvious first step is to take the cube root of both sides.