Exponentials and Logarithms This chapter is focused on functions which are exponential These functions change at an increasing/decreasing rate Logarithms are used to solve problems involving exponential functions

Exponentials and Logarithms Graphs of Exponential Functions You need to be familiar with the function; For example, y = 2 x, y = 5 x and so on… 1) Draw the graph of y = 2 x 3A where /21/2 1/41/4 1/81/8 y x Remember: x y Any graph of will be the same basic shape It always passes through (0,1) as anything to the power 0 is equal to 1

Exponentials and Logarithms Graphs of Exponential Functions Here are a few more examples of graphs where 3A y = 3 x y = 2 x y = 1.5 x All pass through (0,1) They never go below 0 Notice that either side of (0,1), the biggest/smallest values switch Above (0,1), y = 3x is the biggest value, below (0,1), it is the smallest…

Exponentials and Logarithms Graphs of Exponential Functions Here are a few more examples of graphs where 3A y = 2 x y = ( 1 / 2 ) x The graph y = ( 1 / 2 ) x is a reflection of y = 2 x

Exponentials and Logarithms Writing expressions as Logarithms ‘a’ is known as the ‘base’ of the logarithm… 1) Write 2 5 = 32 as a logarithm… 3B means that Effectively, the 2 stays as the ‘first’ number… The 32 and the 5 ‘switch positions’ 2) Write as a logarithm: a)10 3 = 1000 b)5 4 = 625 c)2 10 = 1024

Exponentials and Logarithms Writing expressions as Logarithms 3B means that Find the value of: a) What power do I raise 3 to, to get 81? b) What power do I raise 4 to, to get 0.25? 0.25 is 1 / 4 Remember,

Exponentials and Logarithms Writing expressions as Logarithms 3B means that Find the value of: c) What power do I raise 0.5 to, to get 4? d) What power do I raise ‘a’ to, to get a 5 ? 0.5 = 1 / = 1 / = 4

Exponentials and Logarithms Calculating logarithms on a Calculator On your calculator, you can calculate a logarithm.  Using the log button on the calculator automatically chooses base 10, ie) log20 will work out what power you must raise 10 to, to get 20  To work out log20, all you do is type log20 into the calculator!  log20 = ….  1.30 to 3sf 3C

Exponentials and Logarithms Laws of logarithms You do not need to know proofs of these rules, but you will need to learn and use them: 3D (The Multiplication law) (The Division law) (The Power law) Proof of the first rule: Suppose that; and ‘a must be raised to the power (b+c) to get xy’

Exponentials and Logarithms Laws of logarithms Write each of these as a single logarithm: 3D 1)2)3)

Exponentials and Logarithms Laws of logarithms Write each of these as a single logarithm: 3D 4) Alternatively, using rule 4

Exponentials and Logarithms Laws of logarithms Write in terms of log a x, log a y and log a z 3D 1)2)

Exponentials and Logarithms Laws of logarithms Write in terms of log a x, log a y and log a z 3D 3)4) = 1

Exponentials and Logarithms Solving Equations using Logarithms Logarithms allow you to solve equations where ‘powers’ are involved. You need to be able to solve these by ‘taking logs’ of each side of the equation. All logarithms you use on the calculator will be in base 10. ‘Take logs’ You can bring the power down… Divide by log 10 3 Make sure you use the exact answers to avoid rounding errors.. 3E (3sf)

Exponentials and Logarithms Solving Equations using Logarithms The steps are essentially the same when the power is an expression, such as ‘x – 2’, ‘2x + 4’ etc… There is more rearranging to be done though, as well as factorising. Overall, you are trying to get all the ‘x’s on one side and all the logs on the other… ‘Take logs’ 3E Bring the powers down Multiply out the brackets Rearrange to get ‘x’s together Factorise to isolate the x term Divide by (log7-log3) Be careful when typing it all in! (3dp)

Exponentials and Logarithms Solving Equations using Logarithms You may also need to use a substitution method with even harder ones. You will know to use this when you see a logarithm that has a similar shape to a quadratic equation.. Let y=5 x When you raise a number to a power, the answer cannot be negative… Sub in ‘y = 5 x ’ 3E or y 2 = 5 x x 5 x y 2 = 5 2x Factorise You have 2 possible answers ‘Take logs’ Bring the power down Divide by log5 Make sure it is accurate… (2dp)

Exponentials and Logarithms Changing the base Your calculator will always give you answers for log 10, unless you say otherwise. You need to be able to change the base if your calculator cannot do this You also need to be able to change the base to solve some logarithmic equations 3F Rewrite as an equation ‘Take logs’ to a different base The power law – bring the m down Divide by log b a Sub in log a x for m (from first line)

Exponentials and Logarithms Changing the base 3F (2dp) Special case

Exponentials and Logarithms Changing the base Find the value of log 8 11 to 3.s.f 3F (3sf) Alternatively… ‘Take logs’ Power law Divide by log 10 8

Exponentials and Logarithms Changing the base Solve the equation: log 5 x + 6log x 5 = 5 3F Use the ‘special case’ rule Let log 5 x = y Multiply by y Rearrange like a quadratic Factorise Solve for y

Summary We have learnt what logarithms are We have learnt a number of rules which can be used to manipulate logarithms We have also seen how logarithms can help us solve equations with powers as unknowns