Question

Let u=ln x and v = ln y. Write ln y^7/x^5 in terms of u and v.

Answer 1

`\bf \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad a^(log_a x)=x\qquad \leftarrow \textit{we'll use this one} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ u=\ln(x)\implies u=\log_e(x)\implies e^u=e^(\log_e(x))\implies \boxed{e^u=x} \\\\[-0.35em] ~\dotfill\\\\ v=\ln(y)\implies v=\log_e(y)\implies e^v=e^(\log_e(y))\implies \boxed{e^v=y} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{y^7}{x^5}\implies \cfrac{(e^v)^7}{(e^u)^5}\implies \cfrac{e^(7v)}{e^(5u)}\implies e^(7v)\cdot e^(-5u)\implies e^(7v-5u)`

Answer 2

e^7v-5u

Explanation: