1. The document defines logarithmic functions and their relationship to exponential functions. Logarithmic functions are the inverse of exponential functions with the same base. 2. Methods for determining the logarithm of a number are presented, including using a logarithm table to find the logarithm of numbers that are not perfect powers of their base. 3. The laws of logarithms are proved using properties of exponents, including laws such as log(ab) = log(a) + log(b) and log(a/b) = log(a) - log(b). Examples of applying the laws of logarithms to equations are provided.

1. 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. Review of School Math Content Page 2
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.

3. Review of School Math Content Page 3
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 =

4. Review of School Math Content Page 4
=
= 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 =

5. Review of School Math Content Page 5
=
=
=
=
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 :

6. Review of School Math Content Page 6
–
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 :
= =

7. Review of School Math Content Page 7
= ( ) = ( ) =
4. Logarithm Equation
- with the requirement , so
Example :
Determine the solution of
Answer :
So, the solution of is .
- , with the requirement , so
Example : Determine the solution of
Answer :
So, the solution of is
- with the requirement so .
Example : Determine the solution of Answer :
Now, found that ?




Because for and , and , so and are the completion. So, the solution for is and .

8. Review of School Math Content Page 8
- 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

9. Review of School Math Content Page 9
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)

10. Review of School Math Content Page 10
REFERENCES
Miller, Julie. 2000. Beginning and intermediate AlgebraSecond Edition. California : Benjamin Cummings.
Beyer, William H. CRC Standard Mathematical Tabels. Florida : Universityof Akron.