Solve $ax^t = by^t$ for $ t$

Question

I use to know how to do this but a friend of mine asked for help and I cannot remember.

Can anyone help me solve $ax^t = by^t$ for $t$?

Answer 1

$$ax^t = by^t$$ $$(x/y)^t=b/a$$ $$t\log(x/y)=\log(b/a)$$ $$t=\frac{\log(b/a)}{\log(x/y)}$$

Answer 2

HINT :

Rewrite it as : $$\frac{a}{b} = \Big(\frac{y}{x}\Big)^t$$

Do you know the $\ln$ function? You can use under certain constraints of your constants. Basically need fractions to be positive.

Answer 3

Assuming $x,y>0$ and either $a,b>0$ or $a,b<0$, you can divide by $b$ to obtain $$ \frac ab x^t= y^t$$ Now take logs $$\ln\frac ab+t\ln x = t\ln y.$$ You should be able to solve this for $t$, namely $$t=\frac{\ln\frac ab}{\ln y-\ln x}.$$

If $x,y>0$ and one of $a,b$ is zero or $a,b$ have different signs, there is no solution.

If $x,y>0$ and $a=b=0$ then all $t\in \mathbb R$ are solutions.

If one of $x,y$ is $\le 0$ you may have trouble defining the powers for all $t$ in the first place, but for certain combinations solutions still exist ...