Calculus Workbook: Logarithmic Functions

Section 6.3 Logarithmic Functions

Objectives

Identify logarithmic functions $y=\log_b(x)$ for any base $b >0,b\neq 1$ and its basic properties (domain, range, graphical behavior depending on value of $b$), and relate logarithms to exponentials using inverse relationships
Apply properties and rules of logarithms as well as algebra to: simplify logarithmic expressions, solve equations/inequalities involving exponentials/logarithms, find inverses of exponential/logarithmic functions
Describe the end behavior of logarithmic expressions using limits

Subsection 6.3.1 Before Class

https://mymedia.ou.edu/media/6.3-1/1_d1xy6g44

Figure 52. Pre-Class Video 1

Subsubsection 6.3.1.1 Logarithmic Functions

Definition 6.3.1. Logarithmic Functions.

Let $y = b^x\text{.}$ The inverse of this exponential function, called a logarithmic function, is defined using the relationship $y = b^x \iff x = \log_b(y)\text{.}$

Example 6.3.2.

Find the following:

$\displaystyle \log_4(16)$
$\displaystyle \log_{10} (0.01)$
$\displaystyle \log_3 (243)$
$\displaystyle \log_5 (-5)$

Solution.

$\displaystyle 2$
$\displaystyle -2$
$\displaystyle 5$
Does not exist

Cancellation Properties.

$\log_b (b^x) = x$ for $x\in \R$
$b^{\log_b(x)} = x$ for $x > 0$

Properties of Logarithmic Functions.

Let $f(x) = \log_b(x)\text{.}$ Then, $f(x)$ has the following properties:

Domain: $(0,\infty)$
Range: $(-\infty,\infty)$
If $b\gt 1\text{,}$ then $f(x)$ is increasing
If $0\lt b\lt 1\text{,}$ then $f(x)$ is decreasing
$\displaystyle \ds \lim_{x\to \infty} f(x) = \begin{cases} \infty & \text{if } b\gt 1 \\ -\infty & \text{if } 0\lt b \lt 1 \end{cases}$
$\displaystyle \ds \lim_{x\to -\infty} f(x) = \begin{cases} -\infty & \text{if } b\gt 1 \\ \infty & \text{if } 0\lt b \lt 1 \end{cases}$

Logarithm Rules.

For $x,y\gt 0$ and $r\in\R\text{,}$ the following properties hold:

$\displaystyle \log_b(xy) = \log_b(x) + \log_b(y)$
$\displaystyle \log_b\lrpar{\dfrac{x}{y}} = \log_b(x) - \log_b(y)$
$\displaystyle \log_b(x^r) = r\log_b(x)$

Example 6.3.3.

Use the rules of logarithms to write the following as a single logarithm:

$\displaystyle 2\log_3(x) + 3\log_3(y) - \log_3(z)$
$\displaystyle \log_2(160) + \log_2(10)$

Solution.

$\displaystyle 2\log_3(x) + 3\log_3(y) - \log_3(z) = \log_3\lrpar{\dfrac{xy}{z}}$
$\displaystyle \log_2(160) + \log_2(10) = \log_2(1600)$

Example 6.3.4.

Use the rules of logarithms to expand the given quantity:

$\displaystyle \log_9(\sqrt[3]{ab})$
$\displaystyle \log_{10}\lrpar{\lrpar{\dfrac{x+1}{x-2}}^2}$

Solution.

$\displaystyle \log_9(\sqrt[3]{ab}) = \dfrac{1}{3}\log_9(a) + \dfrac{1}{3}\log_9(b)$
$\displaystyle \log_{10}\lrpar{\lrpar{\dfrac{x+1}{x-2}}^2} = 2\log_{10}(x+1) - 2\log_{10}(x-2)$

Subsection 6.3.2 Pre-Class Activities

Example 6.3.5.

Find the exact value of the expression:

$\displaystyle \log_2(32)$
$\displaystyle \log_{1.5}(2.25)$
$\displaystyle \log_8 (60) - \log_8 (3) - \log_8 (5)$

Solution.

$\displaystyle \log_2(32)=5$
$\displaystyle \log_{1.5}(2.25)=2$
$\displaystyle \log_8 (60) - \log_8 (3) - \log_8 (5) = \dfrac{2}{3}$

Example 6.3.6.

Write $\log_{10}(4) + \log_{10}(a) - \dfrac{1}{3}\log_{10} (a+1)$ as a single logarithm.

Solution.

$\log_{10}(4) + \log_{10}(a) - \dfrac{1}{3}\log_{10} (a+1) = \log_{10}\lrpar{\dfrac{4a}{\sqrt[3]{a+1}}}$

Example 6.3.7.

Can $\log_b(x) + \log_c(y)$ be written as a single logarithm? Why or why not?

Solution.

Not with our current tools; the bases of the logarithms are different

Example 6.3.8.

Let $f(x) = \log_5(8x-x^4)\text{.}$ Use log rules to completely simplify $f(x)\text{,}$ then use limit laws to compute $\ds \lim_{x\to 2^-} f(x)\text{.}$

Solution.

$f(x)$ simplifies as $\log_5(x) +\log_5(x+3) +\log_5(x^2+3x+9)\text{,}$ and the limit is $-\infty$

Question 6.3.9.

Use this space to write any questions or thoughts you have from the videos.

Solution.

Answers vary

Subsection 6.3.3 In Class

Subsubsection 6.3.3.1 The Natural Logarithm

Definition 6.3.10. The Natural Logarithm.

The natural logarithm is the logarithm with base $e\text{.}$ $\log_e(x)$ is written as $\ln x$

Example 6.3.11.

Find $x$ if $e^x=5$

Solution.

$x=\ln 5$

Example 6.3.12.

Sketch the graph of $y=\ln(x-1) + 2$

Solution.

$The graph of $y=\ln(x-1)+2$$

Example 6.3.13.

Solve the equation $e^{3-5x}=10$

Solution.

$x=\dfrac{3-\ln(10)}{5}$

Example 6.3.14.

Solve the equations for $x\text{:}$

$\displaystyle e^{7-4x} = 6$
$\displaystyle \ln(3x-10) = 2$
$\displaystyle \ln(x^2-1) = 3$
$\displaystyle \log_2(mx) = c$
$\displaystyle e-e^{-2x} = 1$
$\displaystyle 10(1+e^{-x})^{-1} = 3$
$\displaystyle e^{2x} - e^x - 6 = 0$

Solution.

$\displaystyle x=\dfrac{7-\ln 6}{4}$
$\displaystyle x=\dfrac{10+e^2}{3}$
$\displaystyle x = \pm \sqrt{e^3+1}$
$\displaystyle x=\dfrac{2^c}{m}$
$\displaystyle x=-\dfrac{\ln(e-1)}{2}$
$\displaystyle x=-\ln\lrpar{\dfrac{7}{3}}$
$\displaystyle x=\ln 3$

Example 6.3.15.

Find the following limits:

$\displaystyle \ds\lim_{x\to 3^+} \ln(x^2-9)$
$\displaystyle \ds\lim_{x\to 2^-} \log_5(8x-x^4)$
$\displaystyle \ds\lim_{x\to 0^+} \ln (\sin x)$

Solution.

$\displaystyle \ds\lim_{x\to 3^+} \ln(x^2-9)=-\infty$
$\displaystyle \ds\lim_{x\to 2^-} \log_5(8x-x^4)=-\infty$
$\displaystyle \ds\lim_{x\to 0^+} \ln (\sin x)=-\infty$

Example 6.3.16.

If $f(x) = \sqrt{3-e^{2x}}\text{,}$ find the domain of $f\text{,}$ the inverse function $\inv{f}\text{,}$ and the domain of $\inv{f}\text{.}$

Solution.

The domain is $\left(-\infty,\dfrac{1}{2}\ln 3\right]\text{,}$ $\inv{f}(x) = \dfrac{1}{2}\ln(3-x^2)\text{,}$ and the domain of $\inv{f}(x)$ is $[0,\infty)$