Indices and Logarithms

  Logarithms are useful for solving problems involving indices (or exponents). In fact, logarithms are just indices in disguise! The definition of a logarithm helps you to see its equivalence with indices: \(a^x=b \quad\Leftrightarrow \quad x=\log _{ a }{ b } \) . Questions about logarithms come up in all different areas of the course so it is important to be comfortable using them. This page will help you with that!


Key Concepts

On this page, you should learn to

  • Convert index equations into logarithms (and vice versa)

\(a^x=b \quad\Leftrightarrow \quad x=\log _{ a }{ b } \)

\(e^x=b \quad\Leftrightarrow \quad x=\ln b\)

  • Use the laws of indices (exponents) and their equivalents for logarithms
Laws of Indices              Laws of Logarithms
\(m^x \times m^y=m^{x+y} \quad \Leftrightarrow \quad\) \(\log _{ c }{ ab }=\log _{ c }{ a } +\log _{ c }{ b } \)
   
\(m^x \div m^y=m^{x-y} \quad \Leftrightarrow \quad\) \(\log _{ c }{ \frac{a}{b} } = \log _{ c }{ a } -\log _{ c }{ b }\)
   
\((m^x)^y=m^{x \times y} \quad \Leftrightarrow \quad\) \(r\log _{ c }{ a }=\log _{ c }{ a ^r}\)
  •  Change the base of a logarithm

\(\log _{ b }{ a } =\frac { \log _{ c }{ a } }{ \log _{ c }{ b } } \)

  • Solve index equations

Essentials

The following videos will help you understand all the concepts from this page

Understanding Logarithms

One of the important things about using logarithms is to be able to see them as indices, but just written in another way \(a^x=b \quad\Leftrightarrow \quad x=\log _{ a }{ b } \)

​Once you understand the conversion, then these questions become more about understanding indices.

In the following video, we will look at these 5 examples

\(1. \quad \log _{ 2 }{ 32 }=x \\ 2. \quad \log _{ 10}{ x }= 3\\ 3. \quad\ln{ 1 }=x \\ 4. \quad\log _{ 5 }{ \frac{1}{125} }=x \\ 5. \quad\log _{ 8}{ \frac{1}{4} } =x\\\)

Notes from the video

Laws of Logarithms

There are laws of indices and these have equivalents for logarithms

Laws of Indices Laws of Logarithms
\(m^x \times m^y=m^{x+y} \quad \Leftrightarrow \quad\) \(\log _{ c }{ ab }=\log _{ c }{ a } +\log _{ c }{ b } \)
   
\(m^x \div m^y=m^{x-y} \quad \Leftrightarrow \quad\) \(\log _{ c }{ \frac{a}{b} } = \log _{ c }{ a } -\log _{ c }{ b }\)
   
\((m^x)^y=m^{x \times y} \quad \Leftrightarrow \quad\) \(r\log _{ c }{ a }=\log _{ c }{ a ^r}\)

​These are useful for solving many different types of problem.

In the following video, we will look at some examples that require us to use these laws:

  1. Show that the equation \(\log _{ 2 }{ (x+2) } +\log _{ 2 }{ (x+3) } =1\) has only one solution and state its value.
  2. Make x the subject of the following formula

\(2+\log _{ 5 }{ x=3\log _{ 5 }{ y } } \)

Notes from the video

Solving Index Equations

In the following video we look at solving equations involving indices. We will solve the following equations:

  1. \(2^x=12\)
  2. \(3^{x+1}=\frac{1}{27}\)
  3. \(5^x=2^{x+1}\)

Notes from the video

Solving Logarithm Equations

Here is a good example of an exam-style question. We are solving an equation that involves both logarithmic and trigonometric functions.

Solve \(\log _{ 4 }{ \frac { cosx }{ 3 } } +\log _{ 4 }{ cosx=-1 } \) , for \(-\pi

Notes from the video

Summary

Print from here

Test Yourself

Here is a quiz that will test your basic understanding of logarithms


START QUIZ!

Here is a quiz about the laws of logarithms


START QUIZ!

Here is a quiz that tests your knowledge of changing bases of logarithms


START QUIZ!

Here is a quiz about solving index equations. You will not need to use logarithms to solve all of these.


START QUIZ!

Here is a quiz that practises ALL the skills from this page


START QUIZ!

Exam-style Questions

Question 1

Find the value of each of the following, giving your answer as an integer

a. \({ log }_{ 4 }16\)

b. \({ log }_{ 4 }2+{ log }_{ 4 }32\)

c. \({ log }_{ 4 }8-{ log }_{ 4 }32\)

Hint

Full Solution

Question 2

a. Given that 3a =27 , write down the value of a

b. Hence of otherwise solve 27x+4 = 93x+1

Hint

Full Solution

Question 3

Solve the equation log2(x - 3) = 1 - log2(x - 4)

Hint

Full Solution

Question 4

Solve the equation \(3^{ x-1 }=\frac { 2 }{ { 4 }^{ x+1 } } \) , giving your answer in the form \(x=\frac{lna}{lnb}\) , where a and b are rational numbers.

Hint

Full Solution

Question 5

Solve \(ln⁡(sinx)−ln⁡(cosx)=e\) for \(0<x<2\pi\)

Hint

Full Solution

 

Question 6

\(a=\log _{ 2 }{ 2+\log _{ 2 }{ \frac { 3 }{ 2 } + } } \log _{ 2 }{ \frac { 4 }{ 3 } + } \quad ...\quad +\log _{ 2 }{ \frac { 32 }{ 31 } } \)

Given that \(a\in \mathbb{Z}\) , find the value of a

Hint

Full Solution

Question 7

The first three terms of a geometric sequence are \(\log _{ 3 }{x\quad ,\quad \log _{ 9 }{ x\quad ,\quad \log _{ 81 }{ x } } } \)

Find the value of x if the sum to infinity is 8.

Hint

Full Solution

MY PROGRESS

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