P3.C2.LOGARITHMS AND EXPONENTIAL - · PDF fileP3.C2.LOGARITHMS AND EXPONENTIAL ! Miss Hjh...

2.1. What is logarithm?

! If a and x are positive real numbers and ! ! , then p is the logarithm of x to the base a, written

! Thus

Any statement in index form such as , has an equivalent logarithmic form, To determine the logarithm to the base a of a number x , we must find the power of a which is equal to x.

! Note:

! Example 1: Convert the following to logarithmic form:

! (a)! (b)! (c)!

!

!! (d)! (e)! (f)

! Example 2: Convert the following to index form:

! (a)! (b)! (c)

! (d)! (e)! (f)

P3.C2.LOGARITHMS AND EXPONENTIAL

! Miss Hjh Rafidah

x = ap

loga x.

x = ap ⇔ p = loga x

25 = 52 log5 25 = 2

1. For to be defined : (a) ! ! ! ! (b)

2. ! and !! and

loga x x > 0a > 0 , a �= 1

loga a = 1 loga 1 = 0 loga an = n

24 = 16 3−2 =1

9100 = 102

a3 = y 2x = p x4 = 2− x

3 = log5 125 −2 = log1

4log4 64 = 3

logx 3 = 4 log3 y = n p+ 1 = log2(4y)

!! Example 3: Check whether the logarithm ! is defined for each of the following:

! (a)! (b)! (c)

! (d)! (e)! (f)

! Example 4: Solve the following equations:

! (a)! (b)! (c)

! (d)! (e)! (f)

! (g)! (h)!


! Miss Hjh Rafidah

logx(5− 2x)

x = 2 x = 0.5 x = 3

x = 2.5 x = 1 x =√2

log2 x = 3 logx 9 = 2 x = log4 8

log3(x− 2) = 1 log2(2x+ 1) = −3 log9√27 = x

logx(6x− 8) = 2 logx 8 =3

2

2.2.! Common and Natural Logarithms

!

Logarithms Definition Example

Common Logarithm:

Logarithm to base 10

- often written as

- can be evaluated using calculator

Natural Logarithm:

Logarithm to base e

-often written as

- can be evaluated using calculator


! (a)! (b)

! Example 6: ! Express in the form Hence find x.


! Miss Hjh Rafidah

log10 or lg

loge or ln

lg Y = X ⇔ Y = 10x

lnY = X ⇔ Y = ex

log10 5 =

log10(x+ 1) =

loge 6 =

loge x =

6x+2 = 21 e3x = 9

3x(22x) = 7(5x) ax = b.


! (a)! (b)! (c)

! (d)! (e)! (f)

! (g)! (h)! (i)

! (j)! (k)! (l)

! Example 8: Solve for x:

!

! (a)! (b)! (c)

! (d)! (e)! (f)

! (g)! (h)

! WORKING:


! Miss Hjh Rafidah

5x = 9 (1.6)x = 21 2(3x) = 5

4− 72x = 1 ex = 7 4e2x = 21

e3x = 14 e4x − 125 = 0 3x+1 = 12

42x−3 = 20 e1+x = 19 (4.1)x = π

lg x = 0.61 (lnx)2 = 3 lnx = lg 2

lg 3x = 9 ln 2 . ln 4x = 3 lg(x− 2) = (lg 3)2

ln 4x = lg 3. lg 5 lg(x+ 1) = ln(e2 − 1)

2.3.! Graphs of Exponential Function .

! (a) if a > 0! (b) if a < 0

! Example 1

! Sketch the following on a graph paper:

! (a)! (b)!

!

! (c)! (d)


! Miss Hjh Rafidah

y = eax

! Example 2:

! By sketching a suitable pair of graphs, show that has only one root.

2.4.! Graphs of Logarithmic Function .

! The function and are inverse functions of each other. Hence the graph of

! is the reflection of the graph of in the line .


! Miss Hjh Rafidah

y = lnx

x = 2 + e−12x

y = lnx y = ex

y = lnx (x > 0) y = ex y = x

! Example 3:

! Sketch the following on a graph paper:

! (a)! (b)!

!

! (c)! (d)

! Example 4: By sketching a suitable pair of graphs, show that equation ! has only one root.


! Miss Hjh Rafidah

2− x = lnx

2.5.! Laws of Logarithms! These rules hold for logarithms to any base a , so the notation has been simplified to

! 1. PRODUCT RULE:

! The logarithm of a product is the sum of the logarithms of the factors.

! Hence,

! Note: The expression is not equal to

! Example 1:

! Simplify the following:

! (a)!

! (b)

! (c)


! Miss Hjh Rafidah

loga x lg x

Product Rule :

Division Rule:

Power Rule :

log(pq) = log p+ log q

log

�p

q

�= log p− log q

log xn = n log x

log(pq) = log p+ log q

log2(3× 5) =

log3 7x =

log4 x(x+ 3) =

log4(x+ 3) log4 x+ log4 3

log6 3 + log6 2

log2 40 + log2 0.1 + log2 0.25

lg 3 + lg 2 + lg 0.3

! 2. DIVISION RULE:

! The logarithm of a division is the logarithm of the numerator minus the logarithm of denominator:

! Hence,

! Note: The expression is not equal to

! Example 2:

! Simplify the following:

! (a)

! (b)

! (c)

! Example 3: Express ! as a single logarithm


! Miss Hjh Rafidah

log

�p

q

�= log p− log q

log3

�7

2

�= log2

�x+ 1

x

�=

log3 7

log3 2log3

�7

2

�

log4 8− log4 2

log2 x4 − log2 x

3

log5 32 − log5 9

1

3log 8− log

2

5

! 3. POWER RULE:

! Hence,

! Note: means . It is not the same as , so

! Example 4: Evaluate the following:

! (a)! (b)

! Example 5: Given that ! and , find

! (a)! (b)

! Example 6: Given that and , express in terms of m and n.


! Miss Hjh Rafidah

log xn = n log x

log2 24 =

log2 x−3 =

log5√x =

loga xn loga(x

n) (loga x)n (loga x)

n �= n loga x

log2 2√2

loga 8

loga 4

loga 2 = 0.301 loga 3 = 0.477

loga3

4loga 2a

loga 3

lg x = m lg y = n lg

�10

�x

y

�

! Example 7: Given that ! where x and y are both positive, express y in

! terms of x.

! Example 8: Express as a single logarithm.

2.6.! Logarithmic Equations! An equation that contains a logarithm of a variable quantity is called a logarithmic equation.

! Logarithmic equations can generally be solved using the following property.

! For example,

! Example 1: Solve the equation

! Example 2: Solve the equation


! Miss Hjh Rafidah

2 lg xy = 2 + lg(1 + x) + lg y

3 + log2 5

For two logarithms of the same base,loga M = loga N ⇔ M = N

log3(x+ 1) = log3 4 ⇔

log2(x− 4) = log2(2x− 6)

log3(x− 1) + log3(x+ 3)− log3(x+ 1) = 1


!

! (a) ! (b)

2.7.! Straight Line Graphs! Non-linear functions in variables x and y can be reduced to linear functions in the form of

! The table below shows how some non-linear functions can be reduced to linear form.

Functions Working Y X m c Graph


! Miss Hjh Rafidah

log3 2 + log3(x+ 4) = 2 log3 x 2 logp 8− logp 4 = 2

Y = mX + cwhere m = gradient

! c = y-intercept

! X and Y are expressions in x and or/ y

(a) y = ax2 + b

(b) y =a

x+ b

(c)1

y= ax2 + b

(d)y = a√x+

b√x

Functions Working Y X m c Graph

! Examples:

1. The diagram shows part of a straight line graph drawn to represent the equation ! . Find the

value of A and n.

2. The table shows experimental values of two quantities x and y which are known to be connected by a law

of the form

x 1 2 3 4

y 30 75 190 470

Plot lg y against x and use your graph to estimate the values of k and b.


! Miss Hjh Rafidah

(e) y = ax2 + bx

(f) y = abx

(g) y = axb

y = Axn

y = kbx.

lg y

lg x

(1, 3)

(2, 1)

3. The table shows experimental values of two variables x and y.

x 0.25 0.30 0.35 0.50 0.60 1.00

y 26.0 18.7 15.6 8.0 5.9 3.5

!It is believed that one of the experimental values of y is abnormally large and also that the variables x and

! y are connected by an equation of the form , where A and B are constants. By a suitable

! choice of variables this equation may be represented by a straight line graph. State these variables and ,

! using the data given above, obtain corresponding pairs of values. Plot these values and hence identify

! the point corresponding to the abnormally large value of y. Ignoring this point, use the remaining points to

! obtain a straight line graph. Use your line to evaluate A and B.

4. Variables x and y are related by the equation . When the graph of lg y against lg x is drawn, the

resulting straight line has a gradient of -2 and an intercept of 0.5 on the axis of lg y. Calculate the values of

p and q.

5. The variables x and y are related in such a way that when is plotted against , a straight line is

obtained passing through (1,-2) and (4,7). Find

! (a) y in terms of x,

! (b) the values of x when y=11.

6. Variables x and y are related by the equation , where p and q are constants. When the

graph of against is drawn, a straight line is obtained. Given that the intercept on the axis is 4.5

and the gradient of the line is -0.8, calculate the value of p and of q.


! Miss Hjh Rafidah

y = A+B

x2

y2 = pxq

y − 2x x2

�x2

p2

�+

�2y2

q2

�

y2 x2 y2

P3.C2.LOGARITHMS AND EXPONENTIAL - · PDF fileP3.C2.LOGARITHMS AND EXPONENTIAL ! Miss Hjh...

Documents

Transcript of P3.C2.LOGARITHMS AND EXPONENTIAL - · PDF fileP3.C2.LOGARITHMS AND EXPONENTIAL ! Miss Hjh...