Chapter 7, Problem 1GYR

To determine

Provide notes on the natural logarithm function defined as an integral, the domain, range, and the derivative of the natural logarithm, and the arithmetic properties of the natural logarithm

Explanation of Solution

The natural logarithm is the function given by $\ln x={\displaystyle {\int }_1^x\frac{1}{t}dt},x>0$.

The number e is the number in the natural logarithm that satisfies $\ln e={\displaystyle {\int }_1^e\frac{1}{t}dt}=1$.

The derivative $\frac{d}{dx}\left(\ln x\right)=\frac{1}{x}$ is positive for $x>0$, so $\ln x$ is an increasing function of x.

The second derivative of $\ln x$ is $-\frac{1}{x^2}$. It is negative, so the graph of $\ln x$ is concave down.

The domain of the function is $\left(0,\infty \right)$.

The range of the function is $\left(-\infty ,\infty \right)$.

Sketch the graph of the function $y=\ln x$ as shown in Figure 1.

Provide the algebraic properties of the natural logarithmic function $\ln x$ as follows.

• $\ln bx=\ln b+\ln x$
• $\ln \frac{b}{x}=\ln b-\ln x$.
• $\ln \frac{1}{x}=-\ln x$.
• $\ln x^r=r\ln x,r\text{ rational}$.