Question

Let u=ln(x) and v=ln(y). Write the expression below in terms of u and v

Answer 1

Answer: (1/2)u - v

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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)\\\\`