How do I find the inverse of an fractional exponential function?

Question

I request that the person who solves this provides steps to explain how they got their answer.

The problem is y = (5^(1+x) + 1)/ (5^x)

I've tried this several times, and I've gotten answers different from the answer key. I'm not sure what I'm doing wrong.

The key says the answer should be log base 1/5 to the power of (x-5). I don't get how the base in the original problem is 5, but the inverse becomes 1/5. Can someone explain how that happens with showing steps?

Answer 1

ln to the base 1/5 is just - ln to the base 5, as (1/5)^n = 5^-n

They are trying to write the answer as simply as possible.

To show the answer, we can show that the inverse transformation is - ln(x-5)/ln(5) - see below. This equals ln(x-5)/ln(1/5). Then, as ln_a b = ln b/ln a, this can be re-expressed as ln_1/5(x-5)

To prove the inverse transformation, why not do this in steps. First, if z = 5^x, then ln z = x ln 5, or x = ln z/ln 5

Then, if y = (5^(1+x) +1)/(5^x), then y = (5z + 1)/z, so zy = 5z+1, or zy-5z = 1 or z(y-5) = 1, or z = 1/(y-5)

Then, as x = ln z/ln 5, x = ln((1/(y-5))/ln 5 = - ln(y-5)/ln(5); thus, the inverse transformation is -ln(x-5)/ln(5)