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17Calculus Precalculus - Logarithms Basics

Single Variable Calculus
Multi-Variable Calculus

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

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.

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

A Special Logarithm - Base \(e\)

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 \( \log_e x \).

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Topics You Need To Understand For This Page

basics of exponentials

basics of logarithms

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Getting Started

What Are Logarithms?

A Special Logarithm


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