3 May 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 3 May 2023 F L1 MH

Change of base ruleIf y = logabThen 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

3 May 2023 F L1 MH

ab

bc

ca log

loglog

Example

Calculate log47 to 3 sig fig

Log47 = log107 / log104 (Change of base)

= Can someone work this out on their Please !

3 May 2023 F L1 MH

..60205.0

..84509.04log7log

7log10

104

A very IMPORTANT result

From the change of base rule we can say

And of course Logyy=1

SO

3 May 2023 F L1 MH

xy

yy

yx log

loglog

xy

yx log

1log

3 May 2023 F L1 MH

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

ALL Pass through (1,0)

3 May 2023 F L1 MH

The inverse to f(x)=logax10(Log10x) is the same as x

And generally

a(Logax) is the same as xSo f(x)= log10(x) and f(x)= 10x are inverse functions. One undoes the other

3 May 2023 F L1 MH

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)=logaxthen 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 ≤33. Measure the gradient at Pt(0,1) carefully4. 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)3 May 2023 F L1 MH

3 May 2023 F L1 MH

3 May 2023 F L1 MH