You CAN Ace Precalculus  

17calculus > precalculus > logarithms  


Topics You Need To Understand For This Page
Precalculus Alpha List
Tools
math tools 

general learning tools 
additional tools 
Related Topics and Links
external links you may find helpful 

Logarithms 

on this page: ► what are logarithms? ► logarithm laws ► graphing logarithms ► next 
The idea of logarithms is not as complicated as it might seem. Logarithms are just another way to write exponents. It's all about notation. The rules that apply to logarithms can be understood if you keep in mind that you are working with exponents. An example should help clarify this. 
You should already be familiar with this expression \(\displaystyle{ 2^3 = 8 }\). This same expression written as a logarithm is
\(\displaystyle{ 3 = \log_2 8 }\) and is read 'three is the logarithm base two of eight' or 'three is log eight base two'.
You could also say, if I have a base two and I want to get eight, what should the exponent of the base two be to get eight? The answer is three.
Let's pause for a minute and watch an interesting video talking about a unique way of looking at exponentials and logarithms.
3Blue1Brown  Triangle of Power [7min44secs]  
It would be nice if all teachers used the triangle idea of thinking about exponentials and logarithms but things won't change overnight. So we need to study and be able to use the traditional way of looking at logarithms.
In calculus, you will work mostly with logarithms with base \(e\). These are special logarithms called natural logarithms. The notation is a bit different. Instead of \( \log_e x \), you will need to write \(\ln(x) \) or \( \ln ~x \). It is considered incorrect notation to write \( \log_e x \).
What Are Logarithms? 
Before we go on, here is a great video for you that explains what logarithms are and how they work. It is well worth your time to watch it.
Dr Chris Tisdell  What are Logarithms? [11min56secs]  
Some Logarithm Laws 
Here are some laws you need to use when combining logarithms.
1. \( \ln(xy) = \ln(x) + \ln(y) \)
2. \( \ln(x/y) = \ln(x)  \ln(y) \)
3. \( \ln(x^y) = y \ln(x) \)
4. \( e^{\ln(x)} = x \)
5. \( \ln(e) = 1 \)
Let's compare the first couple of laws to exponents.
1. \( e^x e^y = e^{x+y} \)
2. \(\displaystyle{ \frac{e^x}{e^y} = e^{xy} }\)
Look at the exponents in these two equations and compare them with the corresponding logarithm law. Do you see the similarities? Spend some time comparing them. Write them next to each other on a piece of paper. Thinking about them and turning them over in your mind repeatedly will help you really understand them and know how to use them.
Here is a good video that proves some logarithm properties. It will help you to understand them and how to use them.
PatrickJMT  A Proof of the Logarithm Properties  
Graphing Logarithms 
Here is a short video discussing how to graph logarithm functions using an example. It is important to have an idea what a logarithm graph looks like. You will need to know this when working with continuity in calculus.
PatrickJMT  Graphing a Logarithm  Made Easy!  
Okay, after working some practice problems, 
next: applications → 
Search 17Calculus
Practice Problems 

Instructions   Unless otherwise instructed, solve these problems using the natural logarithm giving your answers in exact terms.
Level A  Basic 
Practice A01  

Simplify \(\log_2 80\log_2 5\)  
solution 
Practice A02  

Solve \(8^x=15\)  
solution 
Practice A03  

Solve \(1111=5(2^t)\)  
solution 
Practice A04  

Solve \(7^x1=4\)  
solution 
Practice A05  

Solve \(3(2^x)2=13\)  
solution 
Practice A06  

Solve \((2/3)^x=5^{3x}\)  
solution 
Practice A07  

Solve \(5^{x3}=3^{2x+1}\)  
solution 
Level B  Intermediate 
Practice B01  

Solve \(\displaystyle{\left( \frac{4}{5} \right)^x = 6^{1x}}\)  
solution 