Facts About Exponentials and Logarithms

FACTS ABOUT EXPONENTIALS AND LOGARITHMS

I. Definitions of Exponents: We assume m and n are counting numbers.

Definiton: Example:

1. Counting number exponents: x= x 5= 5

if n > 0 then = x ּ x x= x ּ x= x ּ x ּ x ּ x ּ x

2. Zero exponent: = 1 = 1

3. Negative exponent: = =

4. Fractional exponent: = =

= ( ) = ( )

= =

II. Definitions of Logarithm: We assume the base a is a positive number and not 1. These are equivalent:

1. logx is the exponent of a that gives x.

(logx es la potencia a la que debe elevarse a para obtener x.)

2. n = logx means = x

3. = x

4. loga = n

III. Special logarithms:

1. log x = logx So: and

2. ln x = logx where e = 2.71828... So: and

IV. Change of Base for Logarithms:

logx =

V. The THREE GOLDEN RULES:

Exponents: Logs:

1. a ּ a= 1. log= logx + logy

2. = 2. log= logx – logy

3. = 3. log= n (logx)

VI. Functions and Inverses:

Inverses: Examples:

Function
name / Inverse
name / Function / Inverse / Function / Inverse
add / subtract / y = x + a / y = x – a / y = x + 3 / y = x – 3
multiply / divide / y = ax / y = / y = 3x / y =
nth power / nth root / y = x / y = / y = x / y =
exponential / logarithm / y = a / y = logx / y = 2 / y = logx

VII. Forbidden Errors:

1. is NOT equal to xּ xor log (x + y) is NOT equal to log x + log y or log xy

2. is NOT equal to or log (x – y) is NOT equal to log ( ) or

3. is NOT equal to is NOT equal to log x – log y

4. is NOT equal to or (x) (log x)is NOT equal to n (log x)

5. is NOT equal to 0 or 1 log 0 is NOT equal to 1 or 0