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

Lesson 14: Properties of Exponential and Logarithmic Functions

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.

#### Lesson Notes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

## 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 $a>0, a\neq 1)$:
$\log_{b}(x)=\frac{\log_{a}(x)}{\log_{a}(b)}$
```

```
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) $\log _{2}(\sqrt[3]{\frac{x^{2}}{y \sqrt{z}}})$
```

## 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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