Question

For the following, find all the real numbers x that satisfy the equation. If none satisfy the equation then state the answer as "None".
`( (25)/(4) )}^ { - 3x} = (3125)/(32)`(25/4)^-3x = 3125/32

Answer 1

For this case we have the following:

`((25)/(4))^(-3x)=(3125)/(32)`

We can apply natural log in both sides and we have:

`-3x\ln ((25)/(4))=\ln ((3125)/(32))`

And then we can solve for x like this:

`x=-(1)/(3)\cdot(\ln ((3125)/(32)))/(\ln ((25)/(4)))=-(5)/(6)`

Then the solution for this case is :

x= -5/6