TOPIC 1_PART 1_STUDENT

3) loga Mb = b loga M power rule

( loga M ) b b loga M

Remarks:

loga a = 1 loga 1 = 0
ln e = 1 aloga P= P

log 10 = 1

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Example 1 :

Without using calculator,
evaluate of the following :

a) log4 16 b) log5 0.04 c) ln e3

Answer :
a) 2 b)  2 c) 3

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Example 2:

Express each of the following as a single
logarithm.

(a) 1 log 2 9  log2 7  log 2 3
2

(b) 3logc p  n logc q  4logc r

Answer : b logc p3qn
r4
a log2 7

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Example 3:

Express the following in terms of log x, log y
and log z.

a. logxy2  b. log xy  c. log x 
z yz

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Example 4:

Given that log2 3=1.585 and log2 5 = 2.322.
Without using calculator, find the values of

(a) log2 15 (b) log2 1.5

Answer :

a3.907 b 0.585

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Changing The Base Of Logarithms

log a M  log b M log3 5  log 5
log b log
a 2

3

2

loga b  1 a log4 2  log2 2
logb
log12 4
 log2 4
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Example 5: x  log27 x

Given log3 x  m, express log3

in terms of m

The Power of PowerPoint - thepopp.com Answer : m
6

Solving CASE Same base
equations 1
involving
logarithms CASE Same base + have
2 constant

CASE Different base (number)
3

CASE Different base
4 (number + variable)

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CASE  Compare inside log / ln
1 Same base  For more than 2 terms – use law

Example 6: of logarithm
 NEED to check final answer

a) ln3x  lnx  4

b) 2log3 x  log31 x  log32  x Answer :
c) 2ln x  ln 3  ln6  x a. x  2

The Power of PowerPoint - thepopp.com b. x  2
3

c. x  3

CASE Same base +  Collect log/ln term on 1 side
2 have constant  For more than 2 terms – use law

Example 7: of logarithm
 Change log form into index form
a) log2 5  3x  3  NEED to check final answer

b) ln x  2  ln1 x

c) 3log x3  2 log x  2 log x2  8 Answer :

The Power of PowerPoint - thepopp.com a) x  1

b) x  e2
1 e2

c) x  100

CASE Different base  All base are numbers
3 (number)  Change to the lowest number
 NEED to check final answer

Example 8 :

a) log2 2x  log4 x  3

b) log2 x 1 log4 x  1

c) 2 log3 x  log3 x  log9 27 Answer :
a) x 1
The Power of PowerPoint - thepopp.com b) x  1

CASE Different base  Base have number + variable
4 (number + variable)  Change into number
 Use substitution to get quadratic
Example 9:
equation
a) log3 x  log x 9  NO NEED to check final answer

b) log x 3  log3 x6  1

x7 Answer :
2
c) 2 log x 4  log2 a) x  3 2 , x  3 2

1 , x 1

b) x  33 1

32

The Power of PowerPoint - thepopp.com c) x  1 , x  2
256

CASE Same base  Compare inside log / ln
1  For more than 2 terms – use law

CASE Same base + of logarithm
2 have constant  NEED to check final answer

CASE Different base  Collect log/ln term on 1 side
3 (number)  For more than 2 terms – use law

of logarithm
 Change log form into index form
 NEED to check final answer

 All base are numbers
 Change to the lowest number
 NEED to check final answer

CASE Different base  Base have number + variable
4  Change into number

(number + variable)  Use substitution to get quadratic

equation

 NO NEED to check final answer

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ENHANCEMENT EXERCISE :

 1. Solve the equation 22x1  5 2x  3  0

2. Solve the equation 3x  1  2 x  1

3. Solve log2 x  log x 212  7

20/05/2019

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