Question

What is the expression for f(x) when we rewrite (1/32)^x times (1/2)^9x-5 as (1/2)^f(x)

Answer 1

`f(x) = 14x - 5`

Step-by-step explanation

The given expression is

`( (1)/(32))^(x) * ( (1)/(2))^(9x - 5) = ( (1)/(2) ) ^(f(x))`

We rewrite the left hand side using the laws of indices.

`( (1)/(2))^(5x) * ( (1)/(2))^(9x - 5) = ( (1)/(2) ) ^(f(x))`

The bases are now the same on the LHS.

We write one base and add the exponents

`( (1)/(2))^(5x + 9x - 5) = ( (1)/(2) ) ^(f(x))`

`( (1)/(2))^(14x- 5) = ( (1)/(2) ) ^(f(x))`

Since the bases are equal, the exponents are also equal:

`f(x) = 14x - 5`