Unit 4.1: Introduction to Logarithms
Completion requirements
Unit 4
Logarithms
Introduction
Lesson 4.1: Logarithms
Logarithms are closely related to exponents. In Lesson 1.4, only very specific exponential equations were solved algebraically. Learning about logarithms opens the door to solving any exponential equation algebraically.
In Lesson 4.1, you will
- explore the connection between logarithms and exponents,
- graph logarithmic functions,
- use the characteristics of a logarithmic function to evaluate logarithms in various ways, and
- transform the graph of a logarithmic function.
Coming into this Lesson, you should have
- an understanding of the exponent laws,
- the ability to graph exponential functions, and
- an understanding of how a function’s graph is related to its equation.
Math is a field of discovery!
There is always a different way to look at a problem or a different way to solve a problem, and math provides the tools for such exploration.
Did you know that an artist and philosopher named René Descartes was responsible for an entire branch of mathematics involving the use of positive and negative whole numbers?
He wanted to make his paintings more realistic and proportional.
The centre of his paintings are at the intersection point of a horizontal number line and a vertical number line. The location of
an object in his paintings, relative to the origin, or centre, was
defined by its position in terms of coordinates of the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»y«/mi»«/mrow»«/mfenced»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» was
negative to the left of the origin, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» was negative below the origin.
Whoa! A painter made up a new way of thinking about numbers. Few had previously considered the idea of negative values. If negative numbers could exist, then there had to be rules that would apply to these negative numbers.
In a similar way, logarithms is a branch of mathematics that was developed to address a certain need. People such as John Napier from Scotland, who was a physicist among other things, saw a need to work with very large numbers and very small numbers. Back in the 1500s, when Napier was alive, there were no calculators. As such, computations involving very large and/or very small numbers were difficult and time consuming. The advent of logarithms considerably increased the speed of such computations.
Whoa! A painter made up a new way of thinking about numbers. Few had previously considered the idea of negative values. If negative numbers could exist, then there had to be rules that would apply to these negative numbers.
In a similar way, logarithms is a branch of mathematics that was developed to address a certain need. People such as John Napier from Scotland, who was a physicist among other things, saw a need to work with very large numbers and very small numbers. Back in the 1500s, when Napier was alive, there were no calculators. As such, computations involving very large and/or very small numbers were difficult and time consuming. The advent of logarithms considerably increased the speed of such computations.
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