Question

I NEED HELP!! please show all work

What is the exact solution to the equation?
4^(5x)=3^(x-2)

Answer 1

Take the logarithm of both sides. The base of the logarithm doesn't matter.

`4^(5x) = 3^(x-2)`

`\implies \log 4^(5x) = \log 3^(x-2)`

Drop the exponents:

`\implies 5x \log 4 = (x-2) \log 3`

Expand the right side:

`\implies 5x \log 4 = x \log 3 - 2 \log 3`

Move the terms containing x to the left side and factor out x :

`\implies 5x \log 4 - x \log 3 = - 2 \log 3`

`\implies x (5 \log 4 - \log 3) = - 2 \log 3`

Solve for x by dividing boths ides by 5 log(4) - log(3) :

`\implies \boxed{x = -( 2 \log 3 )/( 5 \log 4 - \log 3 )}`

You can stop there, or continue simplifying the solution by using properties of logarithms:

`\implies x = -( \log 3^2 )/( \log 4^5 - \log 3 )`

`\implies x = -( \log 9 )/( \log 1024 - \log 3 )`

`\implies \boxed{x = -\frac{ \log 9 }{ \log \frac{1024}3 }}`

You can condense the solution further using the change-of-base identity,

`\implies \boxed{x = -\log_{\frac{1024}3}9}`