Exponential and logarithmic functions

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Exponential and logarithmic functions illuminated.

Exponential and logarithmic functions may seem somewhat esoteric at first, but they model many phenomena in the real-world.

1 What are exponential and logarithmic functions?

An exponential function is a function of the form

\[ f(x) = b^x \]

where $b\ne 1$ is a positive real number. The domain of an exponential function is $(-\infty ,\infty )$.

Is $b^{-x}$ an exponential function?

yes no

Note that

\[ b^{-x} = \left (b^{-1}\right )^x = \left (\frac {1}{b}\right )^x. \]

A logarithmic function is a function defined as follows

\[ \log _b(x) = y \qquad \text {means that}\qquad b^y = x \]

where $b\ne 1$ is a positive real number. The domain of a logarithmic function is $(0,\infty )$.

In either definition above $b$ is called the base.

1.1 Connections between exponential functions and logarithms

Let $b$ be a positive real number with $b\ne 1$.

$b^{\log _b(x)} = x$ for all positive $x$
$\log _b(b^x) = x$ for all real $x$

What exponent makes the following expression true?

\[ 3^x = e^{\left ( x \cdot \answer {\ln 3} \right )}. \]

2 What can the graphs look like?

2.1 Graphs of exponential functions

Here we see the the graphs of four exponential functions.

Match the curves $A$, $B$, $C$, and $D$ with the functions

\[ e^x, \qquad \left (\frac {1}{2}\right )^{x}, \qquad \left (\frac {1}{3}\right )^{x}, \qquad 2^{x}. \]

One way to solve these problems is to compare these functions along the vertical line $x=1$,

Note

\[ \left (\frac {1}{3}\right )^1 < \left (\frac {1}{2}\right )^1 < 2^1 < e^1. \]

Hence we see:

$\left (\frac {1}{3}\right )^{x}$ corresponds to $\answer [given]{B}$.
$\left (\frac {1}{2}\right )^{x}$ corresponds to $\answer [given]{A}$.
$2^x$ corresponds to $\answer [given]{D}$.
$e^x$ corresponds to $\answer [given]{C}$.

2.2 Graphs of logarithmic functions

Here we see the the graphs of four logarithmic functions.

Match the curves $A$, $B$, $C$, and $D$ with the functions

\[ \ln (x),\qquad \log _{1/2}(x), \qquad \log _{1/3}(x),\qquad \log _2(x). \]

First remember what $\log _b(x)=y$ means:

\[ \log _b(x) = y \qquad \text {means that}\qquad b^y = x. \]

Moreover, $\ln (x) = \log _e(x)$ where $e= 2.71828\dots $. So now examine each of these functions along the horizontal line $y=1$

Note again (this is from the definition of a logarithm)

\[ \left (\frac {1}{3}\right )^1 < \left (\frac {1}{2}\right )^1 < 2^1 < e^1. \]

Hence we see:

$\log _{1/3}(x)$ corresponds to $\answer [given]{B}$.
$\log _{1/2}(x)$ corresponds to $\answer [given]{A}$.
$\log _2(x)$ corresponds to $\answer [given]{D}$.
$\ln (x)$ corresponds to $\answer [given]{C}$.

3 Properties of exponential functions and logarithms

Working with exponential and logarithmic functions is often simplified by applying properties of these functions. These properties will make appearances throughout our work.

3.1 Properties of exponents

Let $b$ be a positive real number with $b\ne 1$.

$b^m\cdot b^n = b^{m+n}$
$b^{-1} = \frac {1}{b}$
$\left (b^m\right )^n = b^{mn}$

What exponent makes the following true?

\[ 2^4 \cdot 2^3 = 2^{\answer {7}} \]

\[ (2^4) \cdot (2^3) = (2 \cdot 2\cdot 2 \cdot 2) \cdot (2 \cdot 2\cdot 2) \]

3.2 Properties of logarithms

Let $b$ be a positive real number with $b\ne 1$.

$\log _b(m\cdot n) = \log _b(m) + \log _b(n)$
$\log _b(m^n) = n\cdot \log _b(m)$
$\log _b\left (\frac {1}{m}\right ) = \log _b(m^{-1}) = -\log _b(m)$
$\log _a(m) = \frac {\log _b(m)}{\log _b(a)}$

What value makes the following expression true?

\[ \log _2\left (\frac {8}{16}\right ) = 3-\answer {4} \]

What makes the following expression true?

\[ \log _3(x) = \frac {\ln (x)}{\answer {\ln (3)}} \]

2025-01-06 19:59:41