EXPONENTIALS & LOGARITHMS

St Andrew's Academy Mathematics Department

Higher Mathematics EXPONENTIALS & LOGARITHMS

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Exponentials and Logarithms

Contents

Exponentials and Logarithms

1 Exponentials 2 Logarithms 3 Laws of Logarithms 4 Exponentials and Logarithms to the Base e 5 Exponential and Logarithmic Equations 6 Graphing with Logarithmic Axes 7 Graph Transformations

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EF 1 EF 3 EF 3 EF 6 EF 7 EF 10 EF 14

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Exponentials and Logarithms

1 Exponentials

EF

We have already met exponential functions in the notes on Functions and Graphs..

A function of the form f ( x ) = ax , where a > 0 is a constant, is known as an exponential function to the base a.

If a > 1 then the graph looks like this: y=y a x , a > 1

(1, a ) 1

This is sometimes called a growth function.

O

x

If 0 < a < 1 then the graph looks like this: y =y ax , 0 < a < 1

1

(1, a )

This is sometimes called a decay function.

O

x

Remember that the graph of an exponential function f ( x ) = ax always

passes through (0,1) and (1, a ) since:

f (0=) a=0 1,

f (1=) a=1 a .

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EXAMPLES

1. The otter population on an island increases by 16% per year. How many full years will it take the population to double?

Let u0 be the initial population.

u1 = 1?16u0 (116% as a decimal)

= u2 1?= 16u1 1?16(1?1= 6u0 ) 1?162u0

( ) = u3 1?= 16u2 1?16 1?16= 2u0 1?163u0

un

=

1?16n

u0

.

For the population to double after n years, we require un 2u0 . We want to know the smallest n which gives 1?16n a value of 2 or more, since this will make un at least twice as big as u0 .

Try values of n until this is satisfied.

I= f n 2, 1?1= 62 1?35 < 2 I= f n 3, 1?1= 63 1?56 < 2 If=n 4, 1?1= 64 1?81 < 2 I= f n 5, 1?1= 65 2?10 > 2

On a calculator:

1 16=

1 1 6 ANS =

=

Therefore after 5 years the population will double.

2. The efficiency of a machine decreases by 5% each year. When the efficiency drops below 75%, the machine needs to be serviced. After how many years will the machine need to be serviced?

Let u0 be the initial efficiency.

u1 = 0?95u0 (95% as a decimal)

= u2 0= ?95u1 0?95(0?9= 5u0 ) 0?952u0

( ) = u3 0= ?95u2 0?95 0?9= 52u0 0?953u0

un = 0?95n u0.

When the efficiency drops below 0?75u0 (75% of the initial value) the machine must be serviced. So the machine needs serviced after n years if 0?95n 0?75.

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Try values of n until this is satisfied: = If n 2, = 0?952 0?903 > 0?75 = If n 3, = 0?953 0?857 > 0?75 = If n 4, = 0?954 0?815 > 0?75 = If n 5, = 0?955 0?774 > 0?75 = If n 6, = 0?956 0?735 < 0?75

Therefore after 6 years, the machine will have to be serviced.

2 Logarithms

EF

Having previously defined what a logarithm is (see the notes on Functions and Graphs) we now look in more detail at the properties of these functions.

The relationship between logarithms and exponentials is expressed as: =y loga x =x a y where a, x > 0 .

Here, y is the power of a which gives x.

EXAMPLES

1. Write 53 = 125 in logarithmic form.

53= 125 3= log5125.

2. Evaluate log416 . The power of 4 which gives 16 is 2, so log416 = 2 .

3 Laws of Logarithms

EF

There are three laws of logarithms which you must know.

Rule 1

loga x + loga y = loga ( xy ) where a, x, y > 0 .

If two logarithmic terms with the same base number (a above) are being added together, then the terms can be combined by multiplying the arguments (x and y above).

EXAMPLE

1. Simplify log5 2 + log5 4 .

log5 2 + log5 4

= log5 (2 ? 4)

= log5 8.

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Rule 2

( ) loga x - loga y = loga xy where a, x, y > 0.

If a logarithmic term is being subtracted from another logarithmic term with the same base number (a above), then the terms can be combined by dividing the arguments (x and y in this case).

Note that the argument which is being taken away (y above) appears on the bottom of the fraction when the two terms are combined.

EXAMPLE

2. Evaluate log4 6 - log4 3 .

log4 6 - log4 3

( ) = log4

6 3

= log4 2

=

1 2

(since 4=12

=4 2).

Rule 3 loga xn = n loga x where a, x > 0.

The power of the argument (n above) can come to the front of the term as a multiplier, and vice-versa.

EXAMPLE

3. Express 2 log7 3 in the form log7 a .

2 log7 3 = log7 32 = log7 9.

Squash, Split and Fly

You may find the following names are a simpler way to remember the laws of logarithms.

? loga x + loga y = loga ( xy ) ? the arguments are squashed together by

multiplying.

( ) ? loga x - loga y = loga xy ? the arguments are split into a fraction.

? loga xn = n loga x ? the power of an argument can fly to the front of the log term and vice-versa.

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Note

When working with logarithms, you should remember:

loga 1 = 0 since a0 = 1,

loga a = 1 since a1 = a .

EXAMPLE

4. Evaluate log7 7 + log3 3 .

log7 7 + log3 3 = 1+1

= 2.

Combining several log terms

When adding and subtracting several log terms in the form loga b , there is a simple way to combine all the terms in one step.

(loga

arguments of positive log terms

)

arguments of negative log terms

Multiply the arguments of the positive log terms in the numerator. Multiply the arguments of the negative log terms in the denominator.

EXAMPLES

5. Evaluate log12 10 + log12 6 - log12 5 .

log12 10 + log12 6 - log12 5

=

log12

10 ? 5

6

= log12 12

= 1.

(log12

)

+ log12 10

+ log12 6

- log12 5

6. Evaluate log6 4 + 2 log6 3 .

log6 4 + 2 log6 3

OR

log6 4 + 2 log6 3

= log6 4 + log6 32

= log6 22 + 2 log6 3

= log6 4 + log6 9

= 2 log6 2 + 2 log6 3

= log6 (4 ? 9)

= log6 36 = 2= (since 62 36).

= 2(log6 2 + log6 3)

= 2(log6 (2 ? 3))

= 2 log6 6

= 2= (since log6 6 1).

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4 Exponentials and Logarithms to the Base e

EF

The constant e is an important number in Mathematics, and occurs

frequently in models of real-life situations. Its value is roughly 2718281828

(to 9 d.p.), and is defined as:

( ) e=

1

+

1 n

n

as n .

If you try very large values of n on your calculator, you will get close to the value of e. Like , e is an irrational number.

Throughout this section, we will use e in expressions of the form: ? e x , which is called an exponential to the base e;

? loge x , which is called a logarithm to the base e. This is also known as the natural logarithm of x, and is often written as ln x (i.e. ln x loge x ).

EXAMPLES

1. Calculate the value of loge 8 . loge 8 = 2.08 (to 2 d.p.). On a calculator: ln 8 =

2. Solve loge x = 9 .

loge x = 9 so x = e9

On a calculator: ex 9 =

x = 8103?08 (to 2 d.p.).

3. Simplify 4 loge (2e ) - 3loge (3e ) expressing your answer in the form a + loge b - loge c where a, b and c are whole numbers.

4 loge (2e ) - 3loge (3e )

OR 4 loge (2e ) - 3loge (3e )

= 4 loge 2 + 4 loge e - 3 loge 3 - 3 loge= e loge (2e )4 - loge (3e )3

= 4 loge 2 + 4 - 3 loge 3 - 3 = 1 + 4 loge 2 - 3 loge 3 = 1 + loge 24 - loge 33 = 1 + loge 16 - loge 27.

=

loge

(2e (3e

)4 )3

=

loge

16e 4 27e 3

Remember

(ab)n = anbn .

=

loge

16e 27

=loge e + loge 16 - loge 27

= 1 + loge 16 - loge 27.

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