Topic 8 Logarithms - University of Adelaide

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(NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014)

Topic 8

Logarithms

y y = ex

5

1

?5

1

0

y = x

?5

y = ln x 5 x

MATHS LEARNING CENTRE

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This Topic...

This topic introduces logarithms and exponential equations. Logarithms are used to solve exponential equations, and so are used along with exponential functions when modelling growth and decay. The logarithmic function is an important mathematical function and you will meet it again if you study calculus. It is used in many areas of advanced applicable mathematics and in statistics.

Author of Topic 8: Paul Andrew

__ Prerequisites ______________________________________________

You will need to have a scientific calculator. We also assume you have read Topic 7: Exponential Functions .

__ Contents __________________________________________________

Chapter 1 Exponential Equations and Logarithms. Chapter 2 Logarithm Functions. Chapter 3 Growth & Decay II. Appendices

A. Answers

Printed: 23/02/2013

1

Exponential Equations & Logarithms

1.1 Exponential Equations

An exponential equation is an equation like 2x = 16 or 10x = 3.267. The first equation has answer x = 4, but the second equation is much harder to solve. An exponential equation has the general form ax = b, where the base a and the number b are known and we wish to find find the unknown index x.

These type of equations arise frequently in growth and decay problems. Example If the population of a town is initially 1000 and is growing at a constant rate of 2% per year, then its population P(t) after t years is given by

P(t) = 1000e0.02t. To find how long it takes for the population to reach 2000, we need to solve the equation

1000e0.02t = 2000 or e0.02t = 2. This equation can be solved once we know that ex = 2 has solution x = 0.6931 (check this), because then we would have 0.02t = 0.6931 => t = 34.66 years.

A logarithm is just an index. We use log as an abbreviation for the word logarithm.

To find the value of a logarithm we need to solve an exponential equation.

Example (a) The solution of 2x = 8 is x = 3.

We can write this in logarithm notation as log 2 8 = 3 (b) x = 5 is the solution of 2x = 32.

`log of 8 to base 2 is 3'

We can write this using logarithms as log 2 32 = 5 (c) 102 = 100.

`log of 32 to base 2 is 5'

We can write this as log 10 100 = 2

`log of 100 to base 10 is 2'

1

Exponentind Logarithms 2

Problems 1.1

1. Rewrite the following in logarithm notation:

(a) 24 = 16

(b) 210 = 1024

(c) 2?1 = 0.5

(d) 20 = 1

(e) 34 = 81

(f) 45 = 1024

(g) 4?0.5 = 0.5

(h) 100 = 1

2. Find the values of the following logarithms:

(a) log 2 4

(b) log 2 16

(e) log 4 4

(f) log 4 16

(c) log 2 1 (g) log 3 1

(d) log 2 0.5 (h) log 10 0.01

1.2 Logarithms

We use can logarithms to solve exponential equations:

The solution of ax = b is x = log a b

For example, the solution of ex = 2 is x = log e 2. To find the value of this logarithm, we need to use a calculator: log e 2 = 0.6931. Note Logarithms were invented and used for solving exponential equations by the Scottish baron John Napier (1550 ? 1617). In those days, before electronic calculators, all logarithms to bases 10 and e were listed in tables. As you can imagine, it was a herculean task constructing these tables of numbers, but the task was made easier because of some properties of logarithms that you'll see later.

Logarithms to the base 10 are called common logarithms. Over their long history, two notations developed: log b (read as `log b') and log 10 b. These both represent the logarithm of b to the base 10.

Logarithms to the base e are used in research, and are called natural logarithms. Once again two notations developed over a long period of time: ln b and log b. These both represent the logarithm of b to the base e.

In most modern texts, including this one, log b refers to the common logarithm of b (base 10), and ln b refers to the natural logarithm of b (base e).

3 Logarithms

Example ? To find the value of ln 2 on a calculator: use ln 2 x=x 0.6931 ? To find the value of log 2 on a calculator: use log 2 x=x 0.3010

Example Solve the equation ex = 3 Answer

ex 3 x ln 3

1.099

Example Solve the equation 10x = 3

Answer 10x 3 x log 3 0.4771

Example

Solve the equation e2x = 3

Answer

e2x 3

2x ln 3

x

ln 3 2

0.5493

Example

Solve the equation e?2x = 3

Answer

e2 x 3

2x ln 3

x

ln 3 (2)

0.5493

Check by calculating e1.099

Check by calculating 100.4771

First find the index using logs, then find x.

Find the index first, then x.

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