Logarithm

Review of School Math Content Page 1

LOGARITHM

1. Definition of a logarithmic function

Consider the following equations in which the variable is located in the exponent

of an expression. In some cases the solution can be found by inspection because the

constant on the right-hand side of the equation is a perfect power of the base.

Equation Solution

= 5 x = 1

= 20 x = ?

= 25 x = 2

= 60 x = ?

= 125 x = 3

The equation = 20 cannot be solved by inspection. However, we might suspect

that x is between 1 and 2. Similary, the solution to the equation = 60 is between 2 and

3. To slove an exponential equation for an unknown exponent, we must use a new type of

function called a logarithmic function.

If x and b are positive real numbers such that b 1, then y = is called the

logarithmic function with base b and

y = is equivalent to

Note : in the expression y = , y is called the logarithm, b is called the base, and

x is called the argument.

The expression y = is equivalent to and indicates that the logarithm

y is the exponent to which b must be raised to obtain x. The expression y = is

called the logarithmic form of the equation, and the expression is called the

exponential form of the equation.

The definition of a logarithmic function suggests a close relationship with

exponential function of the same base. In fact, a logarithmic function is the inverse of the

corresponding exponential function defined by = .

f(x) =

y = replace f(x) by y

x = interchange x and y

y = solve for y using the definition of a logarithmic function

f (x) = replace y by f(x)


2. Determining the logarithm of the numbers

In the previous discussion, you have learned about determining logarithm of the

certain numbers restricted only in the numbers raised to their bases, such as = 3,

because 8 = . The problem is how you can determine the logarithm of a number that is

not resulted from the number raised to it’s base, for examples , , and so on.

One method in determining the logarithm of numbers which are not resulted from the

numbers raised to their bases in using the logarithm table. The table is just for the base

10. Consider a cut of the logarithm table below.

The procedures of reading the logarithm table above are follows

1. If you want to determine log 1,94, the first step is finding that two first

numbers in column N, that is 19

2. The next step is finding decimal (mantis) in the row 19 and column 4 (the

sixth column), you will get value 288

3. Since the value 1,94 is between 1 and 10, so the integer of 1,94 is 0. Thus, we

have log 1,94= 0,288

3. The laws of logarithm

Since the rules and determine the same set of ordered pairs

, the logarithm of the number to the base a is the exponent which must

be used with the base to give .

It is not surprising that the laws of logarithms can be proved by using the laws of

exponents.

is defined for all real numbers y only when , and the value of is

positive for all values of .

Thus is defined only when is positive.


1) = 0

Proof :

= 1

There for = 0

2) = 1

Proof :

There for = 1

3) + =

Proof :

If = m and = n

So, = b and = c

bc = .

bc =

= m+n

= +

There for + =

4) - =

Proof :

If : = m and = n

So, = b and = c

=

=

= m – n

= -

There for -

5) =

Proof :

If : = c =

=

=

b = √

b =


=

= c

Therefor

6) =

Note : The proof of this character uses proof number 5 that has already been

proved before.

Proof :

If : = m = b

= b (then, both of the internodes are given logarithm with base c)

=

=

m =

Therefor =

Of course this character is occurs for c = 10 so =

Applies also to c = b so that =

7) . =

Proof :

. =

.

=

=

Therefor . =

8) = b

Proof :

If : = c

= b

Therefor = b

9) = n

Proof :

If : = v, then =

= x, then =


=

=

=

=

Therefor = n

10)

=

=

=

11) = b

Proof :

From the definition we known

If = c . . . . . . .(1)

So, b = . . . . . . .(2)

By entering c from equation (1) to equation (2) is obtained

b = or = b

Example :

a. =

. determine the value of ?

Answer :

=

=

= 5.

= -

= - 5

b.

. Determine the value of

?

Answer :


–

c. If and , express each of the following problems in terms

of and : a). b). c).

Answer :

a)

=

=

= 1 + 2.

= 1 + 2.

= 1 + 2

b)

=

=

=

c)

=

=

=

=

d. If 9log 8 = n Determine the value of

4log 3 !

Answer :

=

=


=

(

)

=

(

)

=

4. Logarithm Equation

- with the requirement , so

Example :

Determine the solution of

Answer :

So, the solution of is .

- , with the requirement , so

Example :


Answer :

So, the solution of is

- with the requirement

so .

Example :


Answer :

Now, found that ?

Because for and , and , so and are the

completion. So, the solution for is and

.


- with the requirement

, and , so

.

Example :

Answer :

or

Now found that , , , and

Therefore and , or is not the solution.

So, the set of solution for is

-

If so that The value of y substitution to

, so that, we can found the value of x.

Example : Determine the solution of 4log2 - 4log

Answer : 4log2 4log

4log2 – 3. 4log

If , so

y2 – 3y + 2 = 0

(y – 1)(y – 2) = 0

y = 1 or y = 2

to get the value of , subsitution the value of y to ;

, so that

, so that

so, the solution for 4log2 x 4log is

5. Inequality Logarithm

We already know the characteristic of logarithms, as follows:

For , a is an increasing function. Means that for every x1,

x2 if and only if

For , = a is a decreasing function. Means that for every

x1, x2 R x1 < x2 apply x1 < x2 if and only if x1 x2


Example :

1) 4log (

Answer :

4log (

4log

1 4

2

0

(x-1)

Syarat

f(x) 0

0

2 3

1 4

2 3

1 4

2)


REFERENCES

Miller, Julie. 2000. Beginning and intermediate AlgebraSecond Edition. California : Benjamin

Cummings.

Beyer, William H. CRC Standard Mathematical Tabels. Florida : Universityof Akron.

Education

Logarithm