1 Module for Year 10 Secondary School Student Logarithm

2 Module for Year 10 Secondary School Student Logarithm

3 Module for Year 10 Secondary School Student Logarithm

4 Module for Year 10 Secondary School Student Logarithm

Earthquake Intensity Measurement (The Richter Scale)

Dr Charles Richter showed that the larger the energy of an earthquake has, the larger amplitude of ground motion at given distance. The simple model of Richter magnitude using appropriate units is as follow:

Where A is Amplitude ground motion

M is Richter Magnitude

Here some typical effects of earthquakes in various magnitudes range.

Magnitude Effect less than 3,5 Generally not felt, but recorded 3.5 – 5.5 Often felt, but rarely cause damage 5.5 – 6 Slight damage to well-designed building 6.1 – 6.9 Destructive 7.0 – 7.9 Major earthquake 8 or greater Severe earthquake, Severe damage

Charles Richter and Beno Gutenberg develop a model for the relationship between energy and magnitude of resulting earthquake

Where E is energy in Joule

M is the Richter magnitude

푴 = 퐥퐨퐠ퟏퟎ 푨

풍풐품푬 = ퟒퟖ + ퟏ.ퟓ푴

5 Module for Year 10 Secondary School Student Logarithm

What do you expect to learn? This module is designed for you to:

1. Define the logarithmic function f(x) = loga x as the inverse of the exponential function f(x) = ax

2. Understand and apply the law and properties of the logarithm; and 3. Solve simple logarithmic equations.

Motivation and definition The idea of logarithms is to reverse the operation of exponentiation that is raising a number to a power. For example, the third power (or cube) of 4 is, because 64 is the product of three factors of 3:

43 = 4 × 4 × 4 = 64

It follows that the logarithm of 64 with respect to base 4 is 3, so log4 64 = 3.

In contexts to the magnitude of earthquakes in Richter scale in which operates on a logarithmic basis so there is a 10-fold increase from one unit to the next. An earthquake rated 4 on the Richter scale has been assigned a relative intensity of 1 unit. Based on this table, you can learn about the comparison of the magnitude of earthquake.

The Relationship between Richter Scale and Relative Intensity

Richter Scale (RS) 4 5 6 7 8 9 Relative Intensity 1 10 - - - -

What can you say about the Relationship between Richter Scale and relative intensity?

Definition Look at problem below Find the value of n! 10n = 100 100 = …×… n = …. 2n = 32 32 = …. n =….

We can solve the problem above using logarithm. What is the logarithm? From the logarithm definition we can find the value of n 10n = 100 n = log10 102

= log102

n = 2

b= axx = logab (a>0, a ≠ 1, b> 0)

6 Module for Year 10 Secondary School Student Logarithm

2n = 32 n = … n = ….

Some Important Result a. Since 1 = 푎 log 1 = ⋯

Since 푎 = 푎 log 푎 = ⋯

b. log10 is called the common logarithm and just write “log” (without writing base)

c. Natural logarithm written as Ln is logarithm with base e (e = 2. 718281828459)

Let’s do Exercises 1. Write in logarithm form.

a. 125 = 53 b. = 3 c. 32 = 25 Solution

2. Find the value a. log 16 c. log e. log 푎

b. log 4 d. log f. log 10x Solution

The Logarithm The logarithm of a number b with respect to base a is the exponent to which a has to be raised to yield b. In other words, the logarithm of b to base a is the solution x of the equation

b = ax The logarithm is denoted "x = loga(b)" (pronounced as "the logarithm of b to base a" or "the base-a logarithm of b").For the logarithm to be defined, the base a must be a positive real number not equal to 1 and b must be a positive number. Logarithm base 10 is denoted ”log b”

7 Module for Year 10 Secondary School Student Logarithm

Laws of Logarithm

Since

푀 = 푎 and 푁 = 푎 푀.푁 = 푎 .푎 푀.푁 = 푎

From the definition b = ax

x = logab We can write 푀.푁 = 푎 … (The Law I)

Since 푀 = 푎 and 푁 = 푎

= 푎 . 푎 = 푎 From the definition b = ax

x = logab We can write

= 푎 … (The Law II)

Since 푀 = 푎 푀 = (푎 ) From the definition b = ax x = logab

We can write 푀 = 푎 … (The Law III)

Properties of the logarithm

a) ba ba log

b) abb

cc

a logloglog

c) ccb aba log=log×log

d) bnmb a

man loglog

Prove the properties by yourself using definition of logarithm and the law of logarithm!

8 Module for Year 10 Secondary School Student Logarithm

Example: 1. Simplify the following:

a. 24log31log

41log 222

b. 2666 16log64log9log

Solution From the logarithm property: cbcb aaa loglog×log , then

a. 24log31log

41log 222

=

24×

31×

41log2

= 2log2 logaa = 1 = 1

b. 2666 16log64log9log =

......×...log6

= ...log6 = ....

2. a. If p3log5 , then 75log5 = ... (in p form) Solution 75log5 = )3×25(log5 cbcb aaa loglog×log

= 3log25log 55

= p25 5log

= 2 + p

b. If log 9 8 = 3n , then log 4 3 is .... Solution

8log9 = 33 2log 2 3log4 =

4log1

3

3n = 2log......

3 = 2log...1

3

n3......

= 2log3 = ......

9 Module for Year 10 Secondary School Student Logarithm

3. The value of log 4 and log 3 respectively are 0,6021 and 0,4771. Find out the value of log 36 and log 48!

Solution

log 36 = log 4 × 9 log 48 = log ... × ...

= log 4 + log 32 = log ... + log ...

= log 4 + 2. log3 = ...log ... + log ...

= 0,6021 + 0,9541 = ... + ...

= 1,5562 = ....

Logarithmic Equations Logarithmic equations are equations involving logarithmic functions.From previous lesson, you learned that exponent andlogarithmare inverses. Hence, the properties of exponent and logarithm canbe used to solve the simple logarithmic equation Example: Solve the missing terms in the following logarithmic equations.

1. log6216 = x Transform log6216 = x in exponential form, then solve for x. log6216 = x x = ay ↔ y = logax 6x = 216 6x = 63 x = 3

2. log4(5x + 4) = 3

3)45(log4 x

64log)45(log 44 x

64)45( x

12x

3. log(2x + 6) = 2 2)62(log x

...log)62(log x

...)62( x

....x

10 Module for Year 10 Secondary School Student Logarithm

Logarithmic Application for Earthquake Cases

1. How much more intense an earthquake of magnitude 6.5 as compared to one of magnitude 4.5? Solution M1 = log A1 M2 = log A2

6.5 = log A1 4.5 = log A2 A1 = 106.5 A2 = 104.5 So, 푨ퟏ푨ퟐ

= ퟏퟎퟔ.ퟓ

ퟏퟎퟒ.ퟓ

= ퟏퟎퟐ = 100 The movement in the ground for earthquake with magnitude 6.5 is approximate 100 times greater than that of with magnitude 4.5.

2. On 11 January 2012, Meulaboh, Nangroe Aceh Darussalam experienced an earthquake of Richter magnitude 7.3. Calculate the energy associated with this earthquake! Solution

Exercise Fill in the blank

1. 4log25log10log6log 3333 ….

2. If 2364log x , then the value of x = ….

3. 27log319log8log

31

332 = ....

4. 3log.8log.251log 253 ….

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5. If21log 25.0 x , then the value of x = ….

6. bccb log21log

21log

21 ….

7. If 6log4 4 x , then the value of x = ….

8. 31log p

p….

9. There are qp 3log,2log 27 . Transform 98log6 in p and q form is....

10. The value ofx of logarithmic equation 85log 225 x is....

Explain your answer for the following question!

11. How much more intense an earthquake of magnitude 8.4 as compared to the other one of magnitude 6.8?

12. On 11 March 2011, Ishinomaki city Japan experienced an earthquake of Richter magnitude 9.0. Calculate the energy associated with this earthquake!

12 Module for Year 10 Secondary School Student Logarithm