Logarithms - intranet.cesc.vic.edu.au Ticks... · an = x ⇔ log a x = n 1 x 1 8 1 2 1 3 1 2 1 9 1 2 a b 1 2 Logarithms Definition The logarithm of a number is defined by the following:

an = x ⇔ loga x = n

1 x

1 8

1 2

1 3

1 2

1 9

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1 2

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LogarithmsDefinition

The logarithm of a number is defined by the following:

For example 23 = 8 so log2 8 = 3. We say that the log of 8 to base 2 is 3.

Hence the log of a number (x) is the power which the base must be raised to give x.

N.B. loga x is only real when x > 0.

Exercise 1

1). Evaluate the following.

a). log2 4 b). log10 1000 c). log3 3 d). log6 1 e). log3 1 f). log10 0.1 g). log6 36 h). log9 3 i). log5 125 j). log2 16 k). log3 27 l). log4 64 m). log25 5 n). log2 0.5 o). log2 0.25 p) log7 7

q). loga a r). loga a s). logx t). logb (b2)

2). Rewrite the following using index notation.

a). log3 81 = 4 b). log10 2 = 0.301 c). log4 16 = 2 d). log2 ( ) = -3 e). loga 2 = 3 f). log2 y = x g). loga b = 5 h). log4 s = -1 i). logt 4 = 2 j). logx 3 = 4 k). log( ) 32 = p l). logy x = z m). loga 3 = b n). logg h = 7 o). log3 y = d p) logp 6 = q q). loga b = c r). logw q = r s). logi (-1) = 2 t). loge (-1) = i�

3). Rewrite the following using logarithmic notation.

a). 25 = 32 b). 3-1 = c). 102 = 100 d). 360.5 = 6

e). 10-2 = 0.01 f). 72 = 49 g). 4.30 = 1 h). 21 = 2

i). 3-2 = j). ( )-3 = 8 k). a2 = 6 l). b = 38

m). 2n = 5 n). 46.5 = d o). c3 = 30 p). 5g = 1000 q). b3 = 7 r). 2y = x s). st = 4 t). t4 = s

u). sr = f v). pk = e w). qn = x). w-t =

4). Rewrite the following to solve for x.

a). logx 12 = 3 b). log2 x = 8 c). logx 7 = d). log3 x = -1

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b c

b c

x y4

x y4

1 2

1 4

2 15

2 15

5 3

1 4

1 4 2

15 5 3

2 × 5 3

2 15

loga bc = loga b + loga c

loga = loga b – loga c b c

loga (bn) = nloga b

5 3

2 15

5 3

2 15

5 3

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Laws of logs

The laws of indices are reflected in laws for logs.

Let ax = b and ay = c so that ax + y = bc

loga b = x loga c = y loga bc = x + y = loga b + loga c

The above two lines are equivalent statements in index form and logarithmic forms respectively.

1.

Also ax–y = ⇔ loga ( ) = x – y = loga b – logac

2.

Also (ax)n = anx = bn ⇔ loga (bn) = nx = nloga b

3.

Worked Example 1

Express in terms of log x and log y.

log = log ( x ) – log y4 using 2.

= log x – 4 log y using 3.

Worked Example 2

Express log16 – log ( ) + log ( ) as a single logarithm.

log 16 – log ( ) + log ( ) = log 16 – log ( ) + log ( ) using 3.

= log4 16 – log ( ) + log ( )

= log 2 – log ( ) + log ( )

= log using 1. and 2.

= log 25

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1 2

1 2

1 3

1 3

3 2

3 2 1 x – 1

1 3

1 3

5 6

1 3

z x3

x y

1 x

9 x2

yx3 5z2

x3y z

2x y

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Exercise 2

1). Express each of the following as the logarithm of a single number.

a). log10 5 + log10 3 b). log10 3 + log10 7 c). log6 3 + log6 2 d). log10 8 + log10 5 e). log2 7 + log2 5 f). log9 3 + log9 27 g). log2 5 + log2 4 h). log3 12 – log3 4 i). log2 20 – log2 5 j). log3 54 – log3 2 k). log4 128 – log4 8 l). log2 10 – log220 m). 1 – log3 2 n). 3 – log2 9 o). logb x + logb y – logb z

2). Write the following single logarithms.

a) 3log10 2 b). 2log3 5 c). 3log5 2 d) 2log6 10 e). log2 49 f). log3 8

g) 2log10 x h). log2 x i). -2log10 ( )

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

a). log xy b). log c). log (x2y) d). log

e). log (x3 y5) f). log g). log (x5y3z2) h). log 3 x

i). log j). log3 k). log5 l). log2

4). Write the following as single logarithms.

a). 4log 3 + 3log 2 b). 2log 7 + log 2 c). 2log 3 – log 5 d). log 6 – 2log 3 e). 2log x – log y f). log3 6 + 2log3 2 – log3 4

g). log2 125 – log2 49 + 1 h). log(x + 1) + 2log x i). log(x2 – 1) + log

j). 1 + log2 x – log2 y k). 2 + 3log3 x – log3 y l). log5 z – 2log5 y – 2

5). Use log10 2 = p, log10 3 = q and log10 7 = r to find the following in terms of p, q, and r.

a). log10 6 b). log10 8 c). log10 5 d). log10 15 e). log10 1.5 f). log10 0.5 g). log10 9 h). log10 18 i). log10 12.5 j). log10 13.5 k). log10 14 l). log10 30

m). log10 33 n). log10 o). log10 1.8 p). log10 98

6). Solve the following equations correct to 3 significant figures.

a). log3 2x + 2log3 x = 3 b). log6 x + log6 (x + 1) = 1 c). log2 x + 2 = log2 (x + 1)

d). logx 5 = 3 – logx 2 e). log3 x + log3 (x + 3) = 2 f). logx 2 logx 4 =

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log10 6 log10 3

logb b logb a

1 logx 2

1 logx 2

1 y

1 2

1 2

loga x = logb x logb a

loga b = 1 log b a

1 2

logx x logx 2

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Change of base

Let a p = x ⇔ loga x = p 1. b q = x ⇔ logb x = q 2.

Here we have the logs of x to different bases. We can relate these using a relationship between a and b.

Let a = b r ⇔ logb a = r 3.

From 1. a p = x ⇒ (br) p = x so b pr = x ⇒ pr = q

⇒ loga x × logb a = logb x which can be written as :

Worked Example 1

Find log3 6 to 2 decimal places.

We have a log10 x button on the calculator. Using the change of base formula we have

log3 6 = = 1.63

A useful result can be derived from the change of base formula by letting x = b.

loga b = ⇒

Worked Example 2

Solve the equation logx 4 + log2 x = 3.

We can use the change of base formula to write the two log terms to the same base.

The above result will gives: log2 x = = . We can also write logx 4 = logx 22 = 2 logx 2

⇒ logx 4 + log2 x = 2logx 2 + = 3

Now let y = logx 2 so 2y + = 3.

Multiply through by y to get 2y2 + 1 = 3y ⇒ 2y2 – 3y + 1 = 0 ⇒ (2y – 1)(y – 1) = 0

∴ y = 1 or y = ⇒ logx 2 = 1 or logx 2 = ⇒ x1 = 2 or x = 2 ⇒ x = 2 or x = 4.

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1 x

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Exercise 3

1). Use the change of base formula, and the calculator log10 function to find the following, correct to 3 significant figures.

a). log2 7 b). log7 9 c). log5 1.3 d). log20 80 e). log15 6 f). log2 0.1 g). log0.3 25 h). log2 10 i). log70 10

2). Evaluate the following without using a calculator, by changing to base 2.

a). log8 16 b). log512 32 c). log 8 16

d). log 8 e). log 2 ( ) f). log 3 16 2

3). Evaluate the following without using a calculator.

a). log27 9 b). log125 5 c). log 32 8

d). log (7 7 ) e). log 3 f). log0.2 125

g). logx ( ) h). log x i). log x (x2)

j). log(a2)

a k). log (ax)

a l). log n z (zn)

4). If log5 2 = a and log5 3 = b, find the following in terms of a and b.

a). log5 6 b). log5 12 c). log5 10 d). log5 9 e). log5 50f). log5 108 g). log5 0.4 h). log2 6 i). log12 5 j). log0.4 108

5). Show that:

a). 3log27 x = log3 x b). log8 x = log2 x c). loga b = log ( ) ( )


a). log2 x = 9logx 2 b). 6log7 x + 5 logx 7 = 13 c). 2log2 x + 6logx 2 = 7 d). log3 x = log9 (x + 12) e). log2 x + log3 x = 4 f). log3 x = 10 – 6 log9 x

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2 3

2 3

log10 3 log10 2

log10 ( ) log10 2

2 3 2

3

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Index equations

Worked Example 1

Solve the equation 3x = 7.

Taking logs to base 10 of each side we get: log10 3

x = log10 7 ⇒ x log10 3 = log10 7 ∴ x = log10 7 = 1.77 (3 s.f.) log10 3

Worked Example 2

Solve the equation 3 × 22x – 11 × 2x + 6 = 0. Equations of this type can be written as a quadratic equation.

It is important to note that 22x = (2x)2. Substituting y = 2x into the above equation gives:

3 × y2 –11y + 6 = 0 ⇒ (3y – 2)(y – 3) = 0 ⇒ y = 3 or y = ∴ 2x = 3 or 2x = .

If 2x = 3 then x = = 1.58 to 3 s.f.

If 2x = then x = = -0.585 to 3 s.f.

Exercise 4


a). 5x = 7 b). 2x + 1 = 9 c). 101 – x = 80 d). 3x+3 = 2x–2 e). 5x × 3x–1 = 15 f). 102x × 31–x = 7x × 52x+3


a). 6(22x) – 5(2x) + 1 = 0 b). 32x – 5(3x) + 6 = 0 c). 32x – 3x+2 – 10 = 0 d). 2x + 2x+1 + 2x+2 = 49 e). 4x – 2x+2 – 5 = 0 f). 6x+1 – 4x+3 = 80

3). Solve the inequalities.

a). 4x < 25 b). 10x > 600 c). 0.3x < 10-2

4). Solve the simultaneous equations.

2x + 3y = 18

3y+2 – 2x+1 = 118

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x2

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Miscellaneous Exercise

1). If and log2 3 = a and log2 5 = b, find the following in terms of a and b.

a). log2 15 b). log2 10 c). log2 30 d). log2 25 e). log2 9f). log2 50 g). log2 0.4 h). log4 15 i). log15 2 j). log10 15


9y = 32x+3 and log3 y = log3 x + 1.

3). Simplify loga a3 – 2loga ( ).

4). If 2x = 7y find the value of correct to 3 significant figures.

5). Given loga x = p and loga y = q find the following in terms of p and q.

a). loga xy b). loga ( ) c). loga ax d). logx a e) logy x

6). Solve the inequality 0.7x < 0.2, correct to 3 significant figures.


log2 x3 – log2 y = 7

log2 x + log2 y2 = 7

8). Using alogax = x, simplify:

a). 10log104 b). elogecos x c). xlogx3 d). 9log9 3x e). e2loge5

9). Express log10 x2 – 1 + log10 as a single logarithm.

10). Show that logx y4 =

11). Show that logx y × logy z × logz x = 1.

12). Make x the subject of the formula y = 22x.

1 logy (4 x )

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Logarithms - intranet.cesc.vic.edu.au Ticks... · an = x ⇔ log a x = n 1 x 1 8 1 2 1 3 1 2 1 9 1 2 a b 1 2 Logarithms Definition The logarithm of a number is defined by the following: