[PDF] Higher Mathematics EXPONENTIALS & LOGARITHMS

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Mathematics Department

Higher Mathematics

EXPONENTIALS &

LOGARITHMS

1 hsn .uk.net

Exponentials and Logarithms

Contents

Exponentials and Logarithms 1

1 Exponentials EF 1

2 Logarithms EF 3

3 Laws of Logarithms EF 3

4 Exponentials and Logarithms to the Base e EF 6

5 Exponential and Logarithmic Equations EF 7

6 Graphing with Logarithmic Axes EF 10

7 Graph Transformations EF 14

1 hsn .uk.net

Exponentials and Logarithms

1 Exponentials EF

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

Graphs.

A function of the form

x fx a , where 0a is a constant, is known as an exponential function to the base a.

If 1a then the graph looks like this:

This is sometimes called a

growth function. If

01a then the graph looks like this:

This is sometimes called a

decay function. Remember that the graph of an exponential function x fx a always passes through

0,1 and 1,a since:

0

01fa ,

1

1f aa .

O 1 1,a y x , 0 1 x ya a O 1 1,a y x , 1 x yaa 2 hsn .uk.net

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 0 u be the initial population. 10 2

21 0 0

23

32 0 0

0

1·16 (116% as a decimal)

1·16 1·16 1·16 1·16

1·16 1·16 1·16 1·16

1·16 .

nn uu uu u u uu u u uu

For the population to double after

n years, we require 0 2 n uu.

We want to know the smallest

n which gives

1·16

n a value of 2 or more, since this will make n u at least twice as big as 0 u.

Try values of n until this is satisfied.

2 3 4 5

If 2, 1·16 1·35 2

If 3, 1·16 1·56 2

If 4, 1·16 1·81 2

If 5, 1·16 2·10 2n

n n n

On a calculator: 1

1 6 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 0 u be the initial efficiency. 10 2

21 0 0

23

32 0 0

0

0·95 (95% as a decimal)

0·95 0·95 0·95 0·95

0·95 0·95 0·95 0·95

0·95 .

n n uu uu u u uu u u uu

When the efficiency drops below

0

0·75u (75% of the initial value) the

machine must be serviced. So the machine needs serviced after n years if

0·95 0·75

n . 3 hsn .uk.net

Try values of n until this is satisfied:

2 3 4 5 6

If 2, 0·95 0·903 0·75

If 3, 0·95 0·857 0·75

If 4, 0·95 0·815 0·75

If 5, 0·95 0·774 0·75

If 6, 0·95 0·735 0·75n

n n n n 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: log y a y x xa where , 0ax.

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

EXAMPLES

1. Write

3

5 125 in logarithmic form.

3 5

5 125 3 log 125 .

2. Evaluate

4 log 16.

The power of 4 which gives 16 is 2, so

4 log 16 2.

3 Laws of Logarithms EF

There are three laws of logarithms which you must know.

Rule 1

log log log aaa x y xy where ,, 0axy. 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

55
log 2 log 4. 55
5 5 log 2 log 4 log 2 4 log 8. 4 hsn .uk.net

Rule 2

log log log aaa xy xy where ,, 0axy. 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

44
log 6 log 3. 12 44
4 4 63
1 2 log 6 log 3 log log 2 (since 4 4 2).

Rule 3

log log naa xn x where ,0ax.

The power of the

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

EXAMPLE

3. Express

7

2log 3 in the form

7 loga. 7 2 7 7

2log 3

log 3 log 9.

Squash, Split and Fly

You may find the following names are a simpler way to remember the laws of logarithms. log log log aaa x y xy - the arguments are squashed together by multiplying. log log log aaa xy xy- the arguments are split into a fraction. log log naa xn x - the power of an argument can fly to the front of the log term and vice-versa. 5 hsn .uk.net Note When working with logarithms, you should remember: log 1 0 a since 0

1a, log 1

a a since 1 aa.

EXAMPLE

4. Evaluate

73
log 7 log 3. 73
log 7 log 3 11 2.

Combining several log terms

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

1212 12

log 10 log 6 log 5.

1212 12

12 12 log 10 log 6 log 5quotesdbs_dbs12.pdfusesText_18

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