### Home > PC > Chapter 3 > Lesson 3.3.1 > Problem3-106

3-106.

Just below the $\boxed{\text{LOG}}$ key on your calculator is a key labeled $\boxed{\text{LN}}$.

1. Do the three log laws work with $\ln$? For example, is it true that $\ln3+\ln2=\ln\left(3·2\right)$? Check all three laws with several examples before you decide.

Yes, all 3 laws apply.

2. Since LN appears to satisfy all three log laws, it appears to be a log function. LN(X) is written $\ln\left(x\right)$ or $\ln x$ in mathematical notation. When speaking, you say the two letters separately: “el en of $x$” or “el en $x$.” The letter “l” stands for logarithm and the letter “n” stands for natural; $\ln x$ is the natural logarithm of $x$. For logs base $10$, $\log10=1$. For logs base $b$, explain why $\log_bb=1$.

${\text{log}_{\textit{b}}\textit{b} = 1\:\text{because}\:\textit{b}^1 = \textit{b}.}$

3. By experimenting with different inputs, use the result of part (b) to find the base of the natural logarithm function ($\ln$) correct to the nearest $0.001$.

The base of the natural logarithm is close to $x = 2.718$