Question

How to find the x in this exponential function? Thank you!

`2(7^x+2^(x-1))=7(2^x-7^(x-1))`

Answer 1

`log_3.5(2)`

Explanation:

Distributing the variables on each side gives us
`2*7^x+2*2^(x-1) = 7*2^x-7*7^(x-1)`

We know `a^(x-1)*a=a^x` using basic exponent rules, so applying that rule gives us

`2*7^x+2^x=7*2^x-7^x`
Combine like terms
`3*7^x=6*2^x`
`7^x/2^x=2`

Using another exponent rule, we know `a^x/b^2=(a/b)^x`, so
`(7/2)^x=2`
`3.5^x=2`

Using logarithms to solve, we arrive at

`x=log_3.5(2)`