This Means That a Logarithm Is an Index

Logarithms

Definition

/ The logarithm of a number to a particular base is the power (or index) to which that base must be raised to obtain the number.

This means that a logarithm is an index.

The number 8 written in index form is 8 = 23

The equation can be rewritten in

logarithm form as

The logarithm statement reads

Examples

the logarithm of 8, to the base 2 is 3

or 3 is the logarithm of 8 to the base 2

and is equivalent to the index statement

8 equals 2 to the power 3
or 2 to the power 3 equals 8.

In general

1. Change the following from index form to logarithm form.
(a)
the base is 10, the index is 3, the number is 1000
so
(b)
the base is 2, the index is -5, the number is
so
(c)
the base is 16, the index is, the number is 4
so

2. Change the following from logarithm form to index form .

(a)
the base is 2, the number is 16, the index is 4

so (follow the arrows and read)
(b)
the base is 10, the number is 10, the index is 1
so 0

(c)

the base is3, the number is , the index is -4

In general

(follow the arrows and read )

Exercise 1

Write in logarithm form

Write in index form
Evaluating logarithms /

One method for evaluating logarithms is:

let x be the value of the logarithm
rewrite in index form

3. solve the indicial equation.

Example 1

Example 2

/ Evaluate
= x write x as the value of the logarithm

Therefore = 3
Another method is

write the number in index form, with same base as the logarithm
use the definition of a logarithm to evaluate.

Evaluate
125 = 53 125 in index form with base 5
equivalent logarithm statement.
Evaluate
0.0001 = 10-4

Exercise 2 /

Evaluate the following

(a) / (b)
(c)
(e)
(g) / (d)
(f)
(h)
Logarithm laws / The logarithm laws are obtained from the index laws and are:

Note: It is not possible to have the logarithm of a negative number.
All logarithms must have the same base.
Simplifying logarithms / The logarithm laws can be used to simplify logarithmic expressions.
Example 1 / Express the following as a single logarithm.
(Remember the logarithms must have the same base if they are to be added
or subtracted).
(1) Use the laws for adding and subtracting logarithms.

(2) and must be written as and before using addition and subtraction laws.

=
=
=
Example 2 / Simplify and evaluate

=write each term in the form
use laws for adding and subtracting logarithms

Exercise 3 / Express as a single logarithm and evaluate, if possible, without using a calculator .
(a) / (b)
(c) / (d)
(e) / (f)
Solving equations / Equations of the type can be solved for x by rewriting the equation in index form.

Example

/ Solve the following for x.

Therefore x = 32
Logarithms can also be used to solve indicial equations of the form
We solve these equations by taking logarithms of both sides. By taking the logarithms to the base 10 or the base e (Euler’s number), we can use a calculator to evaluate logarithms. Logarithms to the base e are often called natural logarithms.
On the calculator use the log button to evaluate logarithms to the base 10 and the ln button to evaluate logarithms to the base e.
Example / Solve forx giving your answer to three decimal places.
Take logarithms of both sides

Exercise 4 / Solve for the unknown, giving your answer to three decimal places.
(a) / (b)
(c) / (d)
Exponential growth and decay / Many natural processes exhibit exponential growth or decay. Some examples are growth of bacteria, radioactive decay, discharge of a capacitor and rate of temperature change.

Example

/ The number of bacteria present in a sample is given by
, where t is in seconds.
Find:
(a) the initial number of bacteria
(b) the time when the number of bacteria reaches 10 000
(a) The initial number of bacteria occurs when t = 0
Substitute t = 0 in the equation for N. /

The initial number of bacteria is 800.
(b) The number of bacteria, N, is equal to 10 000.
Substitute N = 10 000 in the equationfor N.
Take the logarithm to the base e of both sides. /

It takes 12.6 sec. For the number of bacteria to reach 10 000.
Exercise 5 /

The decay rate for a radio-active element is given by

where R is the decay rate in counts/s at time t(s).
The half–life is the time when R has been reduced to half the initial
value.
If the initial decay rate is 400 counts/s find:
(a) the decay rate after 1 minute
(b) the half-life of the element.
(c) thetime when the decay rate falls to 2 counts/s

Exercise 6

The voltage in a circuit is givenby

If E = 150V, R = 103 ohm, and C = farad, find
The time (seconds) when V = 73V.

Exercise 7

/ The charge Q units on a plate of a condenser t seconds after it
starts to discharge is given by

If Q = 1840 when t = 0.5 and Q = 667 when t = 1 find:
(a ) the value of k
(b) the initial charge Q0
(c)the time taken for the charge to fall to 1000 units.
Graphs of logarithmic functions / All logarithm graphs have the same basic shape.
Consider the graph of .
x and y intercepts
when y = 0 x = 1 i.e the graph cuts the x-axis at the point x = 1.
when x = 0 y is not defined. However,therefore
. the graph has a vertical asymptote at x = 0.
behaviour at

It is not possible to have the logarithm of a negative number, so does not exist at
turning points There are no turning points.
when Therefore the graph passes through the point
Answers

Exercise 1

/ (a) / (b) / (c) / (d)
(e) / (f) / (g) / (h)
(i) / (j) / (k) / (l)

Exercise 2

/ (a) 2 / (b) / (c) 7 / (d) 5
(e) 0 / (f)  3 / (g)  3 / (h)

Exercise 3

/ (a) / (b) / (c) / (d)
(e) / (f)
Exercise 4 / (a) x = 0.699 / (b) x = 11.62 / (c) x = 0.185 / (d) x =  1.807

Exercise 5

/ (a) R = 66 counts/s / (b) t = 23 sec. / (c) t = 177 sec.

Exercise 6

/ t =1 sec.

Exercise 7

/ (a) k = 0.8814 / (b) / (c) t = 0.8 sec.

Created by Sue Thomas/LSU/APS/FELCS;Created on 17/05/2002 9:30 AM Page 1 of 10