1.5 Exponents and Logarithms

  2. Number and Algebra
  3. 1.5 Exponents and Logarithms
    4. 30'
 

Exponents

The exponents of a number tell us how many times the number has been multiplied. Understanding these well, and being able to apply the laws of exponents will help us when sketching graphs, using calculus, and working with very large or very small numbers. 

Key Concepts

In this unit you should learn to…

  • Use the basic laws of exponents
  • Understand the concepts of logarithms
  • Evaluate logarithms and exponents using technology
 

Essentials

Learn about it

The following is a series of videos that will help you understand, learn about and review this sub-topic.

 1. Laws of Exponents

What are the main laws of exponents and why are they useful?

 

 2. Concept of a Logarithm

What are logarithms?

 

3. e and ln

More about the exponential and the natural logarithm

 

4. Using the GDC

A quick look at how to use the GDC to solve problems with exponents and logarithms. 

Summary

 

Test yourself

Quiz

Practice your understanding on these quiz questions. Check your answers when you are done and read the worked solutions when you get stuck. If you find there are still some gaps in your understanding then go back to the videos above.

1

Find

\({ 5 }^{ -2 }\times { 5 }^{ 5 }\)

When the bases are the same, multiply the exponents only.

\({ =5 }^{ (-2+5) }\)

2

Simplify

\({ { (2 }^{ 3 }) }^{ 4 }\)

This is equivalent to \({ { 2 }^{ 3 } } \times { { 2 }^{ 3 } } \times{ { 2 }^{ 3 } } \times { { 2 }^{ 3 } } .\) As the bases are the same we can now simply add together the exponents.

3

Simplify

\(3{ y }^{ -2 } \)

\(3{ y }^{ -2 }=3{ \quad \times \quad y }^{ -2 }\)

\({ y }^{ -2 }=\frac { 1 }{ y^2 } \)

Therefore

\({ 3y }^{ -2 }=3\times \frac { 1 }{ y^2 } \)

4

If \({ a }^{ 3 }\times { a }^{ 4 }={ a }^{ n }\), what is \(n\)?

n =

\({ a }^{ m }\times { a }^{ n }={ a }^{ m+n }\)

5

If \({ ({ 2p }^{ 3 }) }^{ 2 }=a{ p }^{ n }\), what are the values of \(a\) and \(n\)?

a =

n =

\({ ({ 2p }^{ 3 }) }^{ 2 }={ 2 }^{ 2 }\times { p }^{ 3\times 2 }=4{ p }^{ 6 }\)

6

If \({ 5 }^{ -1 } =n\), what is the value of \(n\)? (Without a calculator)

n =

\({ 5 }^{ -1 }=\frac { 1 }{ { 5 } } =0.2\)
7

If \({ ({ 3 }^{ 0 }) }^{ -3 }=\frac { m }{ n } \), what are the values of \(m\) and \(n\)? (Without a calculator)

m =

n =

\({ ({ 3 }^{ 0 }) }^{ -3 }={ 1 }^{ -3 }=\frac { 1 }{ { 1 }^{ 3 } } =\frac { 1 }{ 1 } \)
8

If \({ (\frac { { 9x }^{ 5 } }{ { 4y }^{ 7 } } ) }^{ -3 }=\frac { a{ y }^{ b } }{ m{ x }^{ n } } \), what are the values of \(a\), \(b\), \(m\) and \(n\)?

a =

b =

m =

n =

\({ (\frac { { 9x }^{ 5 } }{ { 4y }^{ 7 } } ) }^{ -3 }=\frac { 1 }{ { (\frac { { 9x }^{ 5 } }{ { 4y }^{ 7 } } ) }^{ 3 } } =1\times { (\frac { { 4y }^{ 7 } }{ { 9x }^{ 5 } } ) }^{ 3 }=\frac { { ({ 4y }^{ 7 }) }^{ 3 } }{ { ({ 9x }^{ 5 }) }^{ 3 } } =\frac { 64{ y }^{ 21 } }{ 729{ x }^{ 15 } } \)
9

If \({ (\frac { { 4a }^{ 3 }\times 2{ a } }{ { 12{ a }^{ 4 }\times a } } ) }^{ -3 }=\frac { { ma }^{ n } }{ x{ a }^{ y } } \), what are the values of \(m\), \(n\), \(x\) and \(y\)?

m =

n =

x =

y =

\({ (\frac { { 4a }^{ 3 }\times 2{ a } }{ { 12{ a }^{ 4 }\times a } } ) }^{ -3 }=\frac { 1 }{ { (\frac { { 4a }^{ 3 }\times 2{ a } }{ { 12{ a }^{ 4 }\times a } } ) }^{ 3 } } =1\times { (\frac { 12{ a }^{ 4 }\times a }{ { { 4a }^{ 3 }\times 2{ a } } } ) }^{ 3 }={ (\frac { { 12a }^{ 5 } }{ { 8{ a }^{ 4 } } } ) }^{ 3 }=\frac { { { ({ 12a }^{ 5 }) }^{ 3 } } }{ { { (8a }^{ 4 }) }^{ 3 } } =\frac { 1728{ a }^{ 15 } }{ 512{ a }^{ 12 } } \)
10

If \(\frac { 5 }{ { a }^{ 4 } } \times { 3a }^{ 8 }=m{ a }^{ n }\), what are the values of \(m\) and \(n\)?

m =

n =

\(\frac { 5 }{ { a }^{ 4 } } \times { 3a }^{ 8 }=5{ a }^{ -4 }\times { 3a }^{ 8 }=15{ a }^{ -4+8 }=15{ a }^{ 4 }\)
 

Exam Style Questions

The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.

Question 1

 

The population (P) of turtles on a beach after t years is modelled by the formula \(P=200\times e^{0.1t}\). 

a. Find the number of turtles: 

i. Initially

ii. After 3 years

b. The year that the population first exceeds 500. 

(4 marks)

Video Solution

 

 Question 2

The Richter scale measures the strength of an earthquake. The strength (S) is given by \(S=\log\left(A\right)\), where A is the amplitude of the wave. 

a. If the amplitude of the wave is 500, find the strength of the earthquake

b. If the strength of the earthquake is 6.5, find the amplitude of the wave

c. If the strength of wave increases by 1, what is the change in the amplitude on the seismograph?

Video Solution

Question 3

Video Solution

Just For Fun

There are some fascinating properties of the number 'e'. You can learn more about them here: 

 
MY PROGRESS

How much of 1.5 Exponents and Logarithms have you understood?