SOLUTION: The instructions say for each function, find f^-1. Help please, step by step. Thank you... f(x)= 6^x-1 Also, solve each equation. Find the Exact solutions. 1/2^x = 5 t

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: The instructions say for each function, find f^-1. Help please, step by step. Thank you... f(x)= 6^x-1 Also, solve each equation. Find the Exact solutions. 1/2^x = 5 t      Log On


   

Question 149457This question is from textbook
: The instructions say for each function, find f^-1. Help please, step by step. Thank you...
f(x)= 6^x-1
Also, solve each equation. Find the Exact solutions.
1/2^x = 5 the exponent is above the 2 only. This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=+6%5E%28x-1%29+ Start with the given function.


x=+6%5E%28f%28x%29-1%29+ Switch x and f(x).


log%2810%2C%28x%29%29=+log%2810%2C%286%5E%28f%28x%29-1%29%29%29 Take the log of both sides.


log%2810%2C%28x%29%29=+%28f%28x%29-1%29%2Alog%2810%2C%286%29%29 Rewrite the right side using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


log%2810%2C%28x%29%29=+f%28x%29%2Alog%2810%2C%286%29%29-log%2810%2C%286%29%29 Distribute


log%2810%2C%28x%29%29%2Blog%2810%2C%286%29%29=+f%28x%29%2Alog%2810%2C%286%29%29 Add log%2810%2C%286%29%29 to both sides.


log%2810%2C%286x%29%29=+f%28x%29%2Alog%2810%2C%286%29%29 Combine the logs using the identity log%28b%2C%28A%29%29%2Blog%28b%2C%28B%29%29=log%28b%2C%28A%2AB%29%29


log%2810%2C%286x%29%29%2Flog%2810%2C%286%29%29=+f%28x%29 Divide both sides by log%2810%2C%286%29%29 to isolate f(x)


log%286%2C%286x%29%29=+f%28x%29 Use the change of base formula to rewrite the left side.


So the inverse function is




%281%29%2F%282%5Ex%29+=+5 Start with the given equation.


1=+5%282%5Ex%29 Multiply both sides by 2%5Ex.


1%2F5=2%5Ex Divide both sides by 5.


log%2810%2C%281%2F5%29%29=log%2810%2C%282%5Ex%29%29 Take the log of both sides.


log%2810%2C%281%2F5%29%29=x%2Alog%2810%2C%282%29%29 Rewrite the right side using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


log%2810%2C%281%2F5%29%29%2Flog%2810%2C%282%29%29=x Divide both sides by log%2810%2C%282%29%29 to isolate x


log%282%2C%281%2F5%29%29=x Use the change of base formula to rewrite the left side.


So the answer is x=log%282%2C%281%2F5%29%29 which approximates to x=-2.32192