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You CAN Ace Precalculus

17calculus > precalculus > logarithms

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Logarithms

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 [7min-44secs]

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? [11min-56secs]
 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. $$\displaystyle{ \frac{e^x}{e^y} = e^{x-y} }$$
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, you will be ready to tackle some application problems involving exponentials and logarithms. next: applications →

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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 $$7^x-1=4$$

solution

Practice A05

Solve $$3(2^x)-2=13$$

solution

Practice A06

Solve $$(2/3)^x=5^{3-x}$$

solution

Practice A07

Solve $$5^{x-3}=3^{2x+1}$$

solution

 Level B - Intermediate

Practice B01

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

solution