158k views
3 votes
Let u=ln(x) and v=ln(y). Write the expression below in terms of u and v

Let u=ln(x) and v=ln(y). Write the expression below in terms of u and v-example-1
User Noobler
by
6.4k points

1 Answer

1 vote

Answer: (1/2)u - v

=======================================================

Work Shown:

We'll apply these log rules


\text{Log Rule 1: } \ \ \ \ln\left((x)/(y)\right) = \ln(x) - \ln(y)\\\\


\text{Log Rule 2: } \ \ \ \ln\left(x^y\right) = y\ln(x)\\\\

These log rules can be used for logs of any base, and not just natural logs.


z = \ln\left((√(x))/(y)\right)\\\\z = \ln\left(√(x)\right)-\ln\left(y\right) \text{ ... Use log rule 1}\\\\z = \ln\left(x^(1/2)\right)-\ln\left(y\right)\\\\z = (1)/(2)\ln\left(x\right)-\ln\left(y\right) \text{ ... Use log rule 2}\\\\z = (1)/(2)u-v\\\\

Therefore,


\ln\left((√(x))/(y)\right) = (1)/(2)u-v\\\\

when
u = \ln(x) \ \text{ and } \ v = \ln(y)\\\\

User Courtnie
by
7.7k points