I would like to remove exponentiation in this equation $y=x^t$ and use only multiplication or division. I have $x, t$ and $\ln(x)$. Is it possible?
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This gives you: $\ln(y) = \ln(x^t) = t \ln(x)\quad\implies \quad \ln(y) = t \ln(x)$
Here we use the rule of logarithms (which applies to "$\ln$" as well):
When $a>0$ then we are always allowed to have $$\ln(a^b)=b\ln(a)$$ So take a Neperian logarithm from both sides of your identity.