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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.

If you want a complete lecture on this topic, we recommend this video from one of our favorite instructors.

Prof Leonard - Exploring the Properties of Logarithms

video by Prof Leonard

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Getting Started

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 (think:exponent) base two of eight' or 'three is log (think:power) eight base two'.
You could also say, if I have a base two and I want to get eight, what should the exponent (or power) 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]

video by 3Blue1Brown

It would be nice if all teachers used the triangle idea when 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]

video by Dr Chris Tisdell

Some Logarithm Laws

Here are some laws you need to use when combining logarithms. We use the natural log here because it is the most common logarithm that you will use in calculus but the same rules apply regardless of the type logarithm.

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 [10min-4secs]

video by PatrickJMT

Okay, let's try some practice problems.

Unless otherwise instructed, simplify these logarithms using the laws above.

Practice 3031
Solution

What is the domain of the logarithm function \( y = \log_2(5-x) + 3 \)?

Problem Statement

What is the domain of the logarithm function \( y = \log_2(5-x) + 3 \)?

Solution

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)		\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)		\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)		\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)		\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)		\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)		\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

\(\int{\sin(x)~dx} = -\cos(x)+C\)		\(\int{\cos(x)~dx} = \sin(x)+C\)
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)		\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)		\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)