More than one way to solve a exponential equation?

Question

What techniques can you use to solve the following equation: $$5 \times 2^x = 2 \times 3^x$$

I know we can use logarithms, but I don't have a lot of confidence solving exponential equations in general. I'm very interested in knowing how one would solve this equation and whether one could use different techniques to do so. I would be very grateful if you could show me how to apply these techniques to this particular problem.

Highly appreciated,

-Bowser

Answer 1

You can transform the equation into $$\frac52 = \left(\frac32\right)^x$$ or $2.5 = 1.5^x$. Equations of the form $a = b^x$ are best solved using logarithms, unless you're very lucky (say $8 = 2^x$ or something). That is not the case here.

Their ability to solve equations like this is half the reason why logarithms even gained popularity back when they were new. The other reason was that you could use them to transform multiplications into additions, which made a lot of practical calculations much faster (although not exact any more).

Answer 2

Let's take the general case, $$a\cdot b^x=c\cdot d^x.$$ We can divide both sides by $d^x$ and $a$, to get: $$\dfrac{b^x}{d^x}=\dfrac{c}{a}\implies\left(\dfrac bd\right)^x=\dfrac ca.$$ Taking the $\log$ of both sides would reveal the solution.