logarithms

19 April 2023 F L1 MH

Objectives :

•To know what log means

•To learn the laws of logs

•To simplify logarithmic expressions

•To solve equations of the type ax=b

1. Simplify the following: 4log4

1910log10b

a alog

(a)

(c)

(b) 1log 2

(d)

(a) 1Ans:

(b) 0

(c) 19

(d) b

Exercises

Change of Base

WE only have log10 and ln (loge)on our calculators

BUT

We can calculate the log to any base logx by rewriting the base

This is called changing the base


Change of base rule

If y = logab

Then ay = bTaking logs of both sides gives logc ay = logcb (c can be any base number)

So ylogc a = logcb ( laws of logs )

So y = logcb/ logca (divide by logca)

Therefore


a

bb

c

ca log

loglog

Example

Calculate log47 to 3 sig fig

Log47 = log107 / log104 (Change of base)

= Can someone work this out on their Please !


..60205.0

..84509.0

4log

7log7log

10

104

A very IMPORTANT result

From the change of base rule we can say

And of course Logyy=1

SO


x

yy

y

yx log

loglog

xy

yx log

1log


red is to base e, green is to base 10, purple is to base 1.

ALL Pass through (1,0)


The inverse to f(x)=logax

10(Log10x) is the same as x

And generally

a(Logax) is the same as x

So f(x)= log10(x) and f(x)= 10x are inverse functions. One undoes the other


Step 1: Let y=logax

Step 2: Rearrange in terms of x

(To do this raise both sides to the power of

a ) ay = alogax

-> ay = xStep 3 : Swap x and y

-> y = ax

If f(x)=logax

then f-1(x) = ax

Exercise - Task

1. Neatly draw the graph of f(x)=ax for these values of a ; 1,2,3. (On graph paper neatly use calculator)

2. Choose your domain to be -4 ≤x ≤3

3. Measure the gradient at Pt(0,1) carefully

4. Guess which value of a gives a gradient of 1 at (0,1)

5. Draw on graph paper f(x)=lnx and ex

6. Try and guess (by considering some points the gradient of ex (at SAY x=-1, 0,1 or x= 0,1,2)


logarithms

Documents

binto foundation l1

otherinto foundation

f l1 mh red

f l1 mh step

f l1 mh objectives

logaxthen f

base e

base logx