3.4 What a logarithm is

Introduction

The base-

The natural logarithm

Let $P(t)$ be the “powers of 10” function, which is given by $P(t) = 10^t$.

Complete the following table to generate certain values of $P$.

$t$ | -3 | -2 | -1 | 0 | 1 | 2 | 3

$y = P(t) = 10^t$ | | | | | | |

$10^t = 0.00001$

$\log_{10}(1000000) = t$

$10^t = 37$

$\log_{10}(y) = 1.375$

$10^t = 0.04$

$3 \cdot 10^t + 11 = 147$

$2\log_{10}(y) + 5 = 1$

What are the domain and range of $E$?

What are the domain and range of $N$?

What can you say about $\ln(e^t)$ for every real number $t$?

What can you say about $e^{\ln(y)}$ for every positive real number $y$?

Complete the following tables with both exact and approximate values of $E$ and $N$. Then, plot the corresponding ordered pairs from each table on the axes provided and connect the points in an intuitive way. When you plot the ordered pairs on the axes, in both cases view the first line of the table as generating values on the horizontal axis and the second line of the table as producing values on the vertical axis. Note that when we take this perspective for plotting the data in the table for $N$, we are viewing $N$ as a function of $t$, writing $N(t) = \ln(t)$ in order to plot the function on the $t$-$y$ axes; label each ordered pair you plot appropriately.

$t$ | $-2$ | $-1$ | $0$ | $1$ | $2$

$E(t)=e^t$ | $e^{-2} \approx 0.135$ | | | |

$y$ | $e^{-2}$ | $e^{-1}$ | $1$ | $e^1$ | $e^2$

$N(y)=\ln(y)$ | $-2$ | | | |