Post on 06-Mar-2018
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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)!
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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
! Example 5: Solve the following equations:
! (a)! (b)
! Example 6: ! Express in the form Hence find x.
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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.
! Example 7: Solve the following equations:
! (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:
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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)
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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 .
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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.
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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)
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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.
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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
! Example 3: Solve the following equations:
!
! (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
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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.
P3.C2.LOGARITHMS AND EXPONENTIAL
! 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.
P3.C2.LOGARITHMS AND EXPONENTIAL
! Miss Hjh Rafidah
y = A+B
x2
y2 = pxq
y − 2x x2
�x2
p2
�+
�2y2
q2
�
y2 x2 y2