Lesson 14: Properties of Exponential and Logarithmic Functions

Hi everyone! Read through the material below, watch the videos, and send your professor your questions.

Topic: This lesson covers Chapter 14 in the book, Exponential and Logarithmic Functions.

WeBWorK: There are three WeBWorK assignments on today’s material: Logarithmic Functions - Properties, Logarithmic Functions - Equations, and Exponential Functions - Equations

Today’s goal is to review the properties/rules of exponents and logs, and then see how we can use them to solve equations.

Properties of Exponential and Logarithmic Functions

You should already be familiar with the rules of exponents and rules of logarithms. I realize you may not remember them, so read through the material below and take a look at the examples in the first video.

Review: Properties of Exponential Functions

The following rules apply to exponential functions (where $b>0$ and $x,y$ are any real numbers):

$b^{x+y}=b^{x} \cdot b^{y}$
$b^{x-y}=\frac{b^{x}}{b^{y}}$
$\left(b^{x}\right)^{n}=b^{n x}$

Review: Properties of Logarithmic Functions

The following rules apply to logarithmic functions (where $b>0, b\neq 1$ and $x,y > 0$, and $n$ is an integer).

$\log_{b}(x \cdot y)=\log_{b}(x)+\log_{b}(y)$
$\log_{b}\left(\frac{x}{y}\right)=\log_{b}(x)-\log_{b}(y)$
$\log_{b}\left(x^{n}\right)=n \cdot \log_{b}(x)$ 

Change of base formula (if $a>0, a\neq 1)$:  
$\log_{b}(x)=\frac{\log_{a}(x)}{\log_{a}(b)}$

Since the logarithm is the inverse of the exponential function, each rule of exponents has a corresponding rule of logarithms.

Example 14.1: Combine the terms using the properties of logarithms so as to write as one logarithm.

a) $\frac{1}{2} \ln (x)+\ln (y)$
b) $\frac{2}{3}\left(\log \left(x^{2} y\right)-\log \left(x y^{2}\right)\right)$
c) $2 \ln (x)-\frac{1}{3} \ln (y)-\frac{7}{5} \ln (z)$ 
d) $5+\log_{2}\left(a^{2}-b^{2}\right)-\log_{2}(a+b)$

Example 14.2:  Write the expressions in terms of elementary logarithms $u=$ $\log_{b}(x), v=\log_{b}(y),$ and, in part $(\mathrm{c}),$ also $w=\log_{b}(z) .$ Assume that $x, y, z>0$

a) $\ln \left(\sqrt{x^{5}} \cdot y^{2}\right)$
b) $\log (\sqrt{\sqrt{x} \cdot y^{3}}) \quad$ 
c) $\log _{2}(\sqrt[3]{\frac{x^{2}}{y \sqrt{z}}})$

Warning: The videos in this lesson are LONG – about 30 minutes each – but they consist almost entirely of EXAMPLES. Feel free to skip around.

Solving Exponential and Log Equations

Now we’re going to use these properties to solve equations.

Example 14.5: Solve for $x$.

a) $2^{x+7}=32$
b) $10^{2 x-8}=0.01$
c) $7^{2 x-3}=7^{5 x+4}$
d) $5^{3 x+1}=25^{4 x-7}$
e) $\ln (3 x-5)=\ln (x-1)$
f) $\log_{2}(x+5)=\log_{2}(x+3)+4$
g) $\log_{6}(x)+\log_{6}(x+4)=\log_{6}(5)$ 
h) $\log_{3}(x-2)+\log_{3}(x+6)=2$

Example 14.6: Solving Log Equations

a) $3^{x+5}=8$
b) $13^{2 x-4}=6$
c) $5^{x-7}=2^{x}$
d) $5.1^{x}=2.7^{2 x+6}$
e) $17^{x-2}=3^{x+4}$
f) $7^{2x+3}=11^{3x-6}$

That’s it for today, everybody! Give the WeBWorK a try.

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