Topic 2 - Logarithmic and Exponential
Topic 2 - Logarithmic and Exponential
Topic 2 - Logarithmic and Exponential
Example 1
Convert the following to logarithm form:
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a) 23 = 8 (b) 3-2 = (c) 2x = 47
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Example 2
Convert the following to exponential form/ index form:
a) log2 32 =5 (b) log3 27 = 3 (c) log2 y = x
Example 3
Find the value of each of the following:
(a) log 2 64 (b) log 3 1 (c) log 7 7 (d) log4 16-1
(e) log3 1 (f) log 8 0.25
81
Law of logarithms.
There are four basic laws of logarithms.
(1) log a mn log a m log a n For two logarithms of the same base,
m loga M = loga N
(2) log a log a m log a n
n
Then, M = N
(3) log a m n n log a m
log c b
(4)log a b
log c a
Example 4
Simplify the following, expressing each as a single logarithm:
(a) log 2 4 + log 2 5 – log210
(b) 2log a 5 – 3 log a 2
(c) log 8 4 + log 2 16
Example 5
If log 2 = r and log 3 = s, express in terms of r and s
(a) log 16
(b) log 18
(c) log 13.5
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Exercise 2.1: Logarithmic and Exponential functions
1. Write each of the following in terms of log p, log q and log r. The logarithms have
base 10.
(a) log pqr
(b) log pq2r3
(c) log 100pr5
p
(d) log
q 2r
pq
(e) log 2
r
1
(f) log
pqr
p
(g) log
r
qr 7 p
(h) log
10
10 p 10 r
(i) log
q
The graphical relationship between y = ax and y = logax is the reflection in the line
y = x.
The functions ex and ln x are inverse functions, the graph of y = ex and y = ln x are
mirror images in the line y = x.
Properties of Indices:
(1) 2a × 2b = 2a + b
(2) 2a ÷ 2b = 2a – b
(3) 2a × 3a = (2×3)a = 6a
(4) (2a)b = 2a× b
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Example 7: Given y = axb and y = 2 when x = 3, y when x = 9, find a and b.
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Example 9: Find the smallest value of n for which the nth term of the geometric
progression with first term 2 and common ratio 0.9 is less than 0.1.
2. The curve y = abx passes through (1, 96), (2, 1152) and (3, p). Find the values of
a, b and p.
3. The curve y = axn passes through (2, 9) and (3, 4). Calculate the values of a and n.
4. Given that y = axb – 5, and that y = 7 when x = 2, and y = 22 when x = 3, find the
values of a and b.
10. How many terms of geometric series 2 + 6 + 18 + 54 + … must be taken for the
sum to exceed 3 million?
11. A biological culture contains 500 000 bacteria at 12 noon on Monday. The culture
increases by 10% every hour. At what time will the culture exceed 4 million
bacteria?
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Properties of logarithms:
Example 11
Solve the equation log2(x – 1) + log2(x + 3) - log2(x + 1) = 1
Example 12
Solve the equation log2 x + 2 log2 x – 2 = 0
Example 13
Solve the simultaneous equation: log2(x – 4y) = 4
log84x – log8(8y + 5) = 1
Example 14
Solve the following logarithmic equations:
(a) log 3 N + log 9 N = 6 (b) log 5 x = 4 log x5
(b) lg x + 2lgy = 3
x2y = 125
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2.6: Linear law: Using logarithm to transform curves into linear lines
Convert
(a) the equation y = axn to logarithmic form, giving a straight line when log y is
plotted against log x.
(b) the equation y = A(bx) to logarithmic form, giving a straight line when log y is
plotted against x.
(c) the equation y = Aenx to the form ln y = ln A + nx, giving a straight line when ln y
is plotted against x.
Example 15
Jack takes out a fixed rate savings bond. This means he makes one payment and
leaves his money for a fixed number of years. The value of his bond, $B, is given by
the formula B = Axn where A is the original investment and n is the number of
complete years since he opened the account. The table gives some values of B and n.
By plotting a suitable graph find the initial value of Jack’s investment and the rate of
interest he is receiving.
n 2 3 5 8 10
B 982 1056 1220 1516 1752
Example 16
The figure shows part of a straight line graph
obtained by plotting values of the variables
indicated. Express y in terms of x.
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Exercise 2.5: Using logarithm to transform curves into linear lines
4. The table shows the mean relative distance, X, of some of the planets from the
Earth and the time, T years, taken for one revolution round the sun. By drawing an
appropriate graph show that there is an approximate law of the form T = aX n,
stating the values of a and n.
Mercury Venus Earth Mars Saturn
X 0.39 0.72 1.00 1.52 9.54
T 0.24 0.62 1.00 1.88 29.5