Logarithms

AS Physics Skills

Homework

Complete logarithms homework sheet.

Learning ObjectivesLearning Objectives

To learn how to use logarithms to To learn how to use logarithms to solve equations.solve equations.

LogarithmsLogarithms

100 = 10100 = 1022

In this statement we say that 10 is the In this statement we say that 10 is the base base and 2 is the power or index. and 2 is the power or index.

Logarithms provide Logarithms provide an alternative way of an alternative way of writing a statement such as this. We rewrite writing a statement such as this. We rewrite it asit as

loglog1010 100 = 2 100 = 2

This is read as ‘log to the base 10 of 100 is 2’. This is read as ‘log to the base 10 of 100 is 2’.

LogarithmsLogarithms

LogarithmsLogarithms

I like to think of logI like to think of logbba as meaninga as meaning “ “what power of b is a?”what power of b is a?”

So logSo log101010000 translates to:-10000 translates to:- ““what power of 10 is 10,000?” what power of 10 is 10,000?” =4 =4

So logSo log3327 translates to:-27 translates to:- ““what power of 3 is 27?” what power of 3 is 27?” =3 =3

Another ExampleAnother Example

2255 = 32 = 32

we can write this aswe can write this as loglog22 32= 5 32= 5

Here the base is 2 and the power is Here the base is 2 and the power is 5. We read this as ‘log to the base 2 5. We read this as ‘log to the base 2 of 32 is 5’.of 32 is 5’.

ee

e is a special number, a bit like e is a special number, a bit like ππ

It has the property that if you plot y=eIt has the property that if you plot y=exx, , then at every point on the curve the slope then at every point on the curve the slope also equals the y-valuealso equals the y-value

For example, if x=5, then y=eFor example, if x=5, then y=e55 and the and the slope at that point, =eslope at that point, =e55..

e = 2.718281828e = 2.718281828

Log Rule 1Log Rule 1

mnnm bbb logloglog

3log2log32log bbb

3log2log6log bbb

Example with Numbers:-

Log Rule 2Log Rule 2

Examples with Numbers:-Examples with Numbers:-

mnm

nbbb logloglog

4log12log4

12log bbb

4log12log3log bbb

Log Rule 3Log Rule 3

Examples with Numbers:-Examples with Numbers:-

m log p m log bp

b

4 log 5 4 log b5

b

9 log 6 9 log b6

b

Log RulesLog Rules

Examples with Numbers:-Examples with Numbers:-

p blog pb

3 10log 310

24 8log 248

Inverse LogsInverse Logs

Examples with numbers:-Examples with numbers:-

m b m logb

600 10 600 log10 23 7 23 log7

Standard BasesStandard Bases

In Science, we tend to use only two In Science, we tend to use only two bases either log to the base 10, bases either log to the base 10, which is written as just “log” or as which is written as just “log” or as “lg”.“lg”.

Or we use log to the base e (natural Or we use log to the base e (natural logarithm), which is written as “ln”.logarithm), which is written as “ln”.

Similarly,Similarly, x 10 xlg x 10 lg x

x eln x x eln x

Solving EquationsSolving Equations

For example, say we want to find x For example, say we want to find x for:-for:-

Log both sides:-Log both sides:-

Using log rule No.3:-Using log rule No.3:-

Re-arranging:-Re-arranging:-

5 3x

5 log 3 log x

5 log 3 logx

3 log

5 log x