Question

Solve the equation 2(3^x) = 3(4^x). Give an exact answer and a decimal approximation rounded to one decimal place.

Answer 1

I think I have to use logs to solve this

Answer 2

first off: moving the alike terms to one side
\[2(3^x)=3(4^x)\\\dfrac{1}{3}3^x=\dfrac{1}{2}4^x\]
observe:
\(\dfrac{1}{3}= 3^{-1}\) and \(\dfrac{1}{2} = 4^{-1/2}\)
so that your equation become
\[3^{-1}3^x=4^{-1/2}4^x\\3^{x-1}=4^{x-1/2}\]
can you handle it from here?

Answer 3

this is different as to how I was going to handle It, I thought the first step would be to divide both sides by 2 and then solve using logarithms

Answer 4

of course you have to use log to solve,
now take ln both sides to get
\[ln(3^{x-1})= ln(4^{x-1/2})\\(x-1) ln 3 = (x-1/2)ln4\]
so that
\[\dfrac{x-1}{x-1/2}=log_3 4\]
now solve for x