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Author: Jonas Reitz (Page 2 of 2)

Lesson 14: Properties of Exponential and Logarithmic Functions

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)}

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.

Lesson 13: Exponential and Logarithmic Functions

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

Lesson 13: Exponential and Logarithmic Functions

Topic: This lesson covers Chapter 13 in the book, Exponential and Logarithmic Functions.

WeBWorK: There are two WeBWorK assignments on today’s material: Exponential Functions - Graphs and Logarithmic Functions - Graphs .

Lesson Notes (Notability – pdf):

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

Exponential Functions and their Graphs

We’ve been living in the world of Polynomials and Rational Functions. We now turn to exponential functions. These functions are “very natural” – that is, they show up in the real world – but they are also more complicated than Polynomial and Rational functions (for example, an exponential function grows more quickly than any Polynomial)

The spread of coronavirus, like other infectious diseases, can be modeled by exponential functions.

Definition. An exponential function is a function of the form f(x)=c\cdot b^x, where b and c are real numbers and b is positive (b is called the base, x is the exponent).

Example 1 (Textbook 13.2): Graph the exponential functions f(x)=2^x, g(x)=3^x, h(x)=10^x, k(x)=\left(\frac{1}{2}\right)^x, l(x)=\left(\frac{1}{10}\right)^x.

Now let’s see what happens when we change the number c in y=c\cdot b^x.

Example 2 (Textbook 13.6): Graph the exponential functions
a) y=2^{x}, \quad b) y=3 \cdot 2^{x}, \quad c) y=(-3) \cdot 2^{x}, \quad d) y=0.2 \cdot 2^{x}, \quad e) y=(-0.2) \cdot 2^{x}

Example 3: The graph below shows an exponential function f(x). Find a formula for f(x).

Logarithmic Functions and their Graphs

Definition. If b is a positive real number and b\neq 1, then the logarithm with base b is defined:
y=\log_b(x) \iff b^y=x

What does the definition of logarithm mean? The idea is that the logarithm is the inverse function of the exponential function. Let’s look at an example.

Question: Is an exponential function one-to-one? (What does one-to-one means).

Example 4. The graph below shows the function y=\log_2(x) but shifted to the right 3 units. Find a formula for the function in the graph.

That’s it for now! Take a look at the WeBWorK assignment, leave your questions below (or use the Ask for Help button in WeBWorK, or send me an email)

Lesson 12: Polynomial and Rational Inequalities

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

Lesson 12: Polynomial and Rational Inequalities

Topic: This lesson covers Chapter 12 in the book, Polynomial and Rational Inequalities.

WeBWorK: There are two WeBWorK assignments on today’s material: Polynomials - Inequalities, and Rational Functions - Inequalities.

Introduction to polynomial inequalities

Definition. A polynomial inequality is an inequality (which means it uses one of these: <,\leq,>,\geq instead of an equals sign) with a polynomial on each side

Example 1: x^2-3x-4\geq 0

We’re interested in solving these inequalities, which means answering the question: “For which real numbers x is the inequality true?”

Now let’s look at the same example, and see how to solve it without looking at the graph:

Example 2: Solve x^{4}-x^{2}>5\left(x^{3}-x\right)

Example 2, concluded:

Rational inequalities

What happens if we allow rational functions instead of just polynomials?

Example 3: Solve \frac{x^{2}-5 x+6}{x^{2}-5 x} \geq 0

Good job! You are now ready to practice on your own. Take a look at the WeBWorK assignment, and don’t forget to use the “Ask for Help” button if you get stuck.

Here are more video resources if you’d like to see additional examples.

ASSIGNMENT: Watch videos, try webwork.

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This site collects resources and information about the course MAT 1375 Precalculus.

If you are a current or future student in MAT 1375, you will find basic course information (like the syllabus), online lessons for each day of the course, and links to support materials like review sheets, tutoring information, WeBWorK help, and more.

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