1. Exponential and Logarithmic Functions
Section 4.2
Exponential and
Logarithmic Functions

2. LAWS OF EXPONENTS
Laws of Exponents with General Base a: If the base number a is positive and x and y are any real numbers, then

3. ADDITIONAL EXPONENT LAWS

4. FRACTIONAL EXPONENTS
Recall that radicals can be expressed as fractional exponents. That is,
Below are some examples.

5. LAWS OF EXPONENT WITH BASE e
If x and y are real numbers, then

6. COMMON LOGARITHMS
Definition: The common logarithm of the positive number x is the power to which 10 must be raised in order to obtain the number x. It is denoted by log10 x. Thus,
y = log10 x means the 10y = x.
Frequently, we omit the subscript 10 and simply write log x for the common logarithm of the positive number x.

7. NATURAL LOGARITHMS
Definition: The natural logarithm of the positive number x is the power to which e must be raised in order to obtain the number x. It is occasionally denoted by loge x, but more frequently by ln x (with l for “log” and n for “natural”). Thus,
y = ln x means that ey = x.
NOTE: Only positive numbers have logarithms (common or natural).

8. LAWS OF LOGARITHMS
Laws of Logarithms: If x and y are positive real numbers, then
The logarithm of a product is the sum of the logarithms.
The logarithm of a quotient is the difference of the logarithms.
The logarithm of a reciprocal is the negative of the logarithm.
The logarithm of a power is the exponent times the logarithm of the base.
The logarithm of one is zero.

9. EXPONENTS AND LOGARITHMS AS INVERSES
Just as addition and subtraction (and multiplication and division) undo each other, exponentials and logarithms undo each other also. That is,
eln x = x and ln ex = x.
Two functions, that undo each other are called inverses.