Chapter 3.3, Problem 73E

To determine

To condense the expression $\frac{1}{2}\left[2\ln \left(x+3\right)+\ln x-\ln \left(x^2-1\right)\right]$ to the logarithm of a single quantity.

Answer to Problem 73E

  $\frac{1}{2}\left[2\ln \left(x+3\right)+\ln x-\ln \left(x^2-1\right)\right]=\ln {\left({\left(x+3\right)}^2\times \frac{x}{\left(x^2-1\right)}\right)}^{\frac{1}{2}}$

Explanation of Solution

Given information:

Consider the expression $\frac{1}{2}\left[2\ln \left(x+3\right)+\ln x-\ln \left(x^2-1\right)\right]$

Calculations:

Use Quotient property for Natural Logarithm, that is., $\ln \frac{u}{v}=\ln u-\ln v$ where u and v are positive real numbers.

  $\Rightarrow \frac{1}{2}\left[2\ln \left(x+3\right)+\ln \frac{x}{\left(x^2-1\right)}\right]$

Next, apply power property for Natural Logarithm, $\ln u^n=n\ln u$ where n is a real number and u is a positive real number.

  $\Rightarrow \frac{1}{2}\left[\ln {\left(x+3\right)}^2+\ln \frac{x}{\left(x^2-1\right)}\right]$

Now use Product property for Natural Logarithm, that is., $\ln \left(uv\right)=\ln u+\ln v$ where u and v are positive real numbers.

  $\Rightarrow \frac{1}{2}\left[\ln \left({\left(x+3\right)}^2\times \frac{x}{\left(x^2-1\right)}\right)\right]$

Again, apply power property for Natural Logarithm and get the final result:

  $\Rightarrow \ln {\left({\left(x+3\right)}^2\times \frac{x}{\left(x^2-1\right)}\right)}^{\frac{1}{2}}$

Hence, condensed form of the expression: $\frac{1}{2}\left[2\ln \left(x+3\right)+\ln x-\ln \left(x^2-1\right)\right]=\ln {\left({\left(x+3\right)}^2\times \frac{x}{\left(x^2-1\right)}\right)}^{\frac{1}{2}}$