Chapter 3, Problem 70RE

To determine

To expand the expression

  $\ln \frac{\sqrt{x}}{4}$

Answer to Problem 70RE

  $\ln \frac{\sqrt{x}}{4}=\frac{1}{2}\ln x+\ln 4$

Explanation of Solution

Given:

  $\ln \frac{\sqrt{x}}{4}$

Concept Used:

• Power property of logarithmic function: $\ln u^n=n\ln u$
• Quotient property of logarithmic function: $\ln \frac{u}{v}=\ln u-\ln v$

Calculation:

To expand the expression

  $\ln \frac{\sqrt{x}}{4}$

Simplify this using the properties of logarithmic function $\ln \frac{u}{v}=\ln u-\ln v$.

As,

  $\begin{array}{c}\ln \frac{\sqrt{x}}{4}=\ln \sqrt{x}-\ln 4\\ =\ln x^{1/2}-\ln 4\end{array}$

Now to further expand use the property $\ln u^n=n\ln u$.

As,

  $\begin{array}{c}\ln \frac{\sqrt{x}}{4}=\ln x^{1/2}-\ln 4\\ =\frac{1}{2}\ln x+\ln 4\end{array}$

Thus,

  $\ln \frac{\sqrt{x}}{4}=\frac{1}{2}\ln x+\ln 4$