A logarithm is an exponent. A logarithm is an exponent which indicates to what power a base must be raised to produce a given number.

*y
= b ^{x}*
exponential form

*x
= *log* _{b} y* logarithmic
form

x is the logarithm of y to the base b

log_{b}y is the power to which we have to raise b to get y

We are expressing x in terms of y

**Examples**

**x = ****log_{b} y**

x =log_{2}8This means the logarithm of 8 to the base 2. It is the exponent to which 2 must be raised to get 8. We know that 2(2)(2) = 8. Therefore x = 3.

x =log_{6}36This means the logarithm of 36 to the base 6. It is the exponent to which 6 must be raised to get 36. We know that 6(6) = 36. Therefore x = 2.

x =log_{10}10,000This means the logarithm of 10,000 to the base 10. It is the exponent to which 10 must be raised to get 10,000. We know that 10(10)(10)(10) = 10,000. Therefore x = 4.

log _{b}b = 1The logarithm of any number to the same base equals 1.

x = log _{11}11This means the logarithm of 11 to the base 11. It is the exponent to which 11 must be raised to get 11. We know that 1 (1) = 11. Therefore x = 1.

log _{b}1 = 0The logarithm of 1 always equals 0.

Any number can serve as b, the base.

Common (Briggsian) logarithms The base is 10.Logarithms to the base 10 are widely used. Thus it is common to drop the subscript. If the base does not appear it is understood that the base is 10.log

_{10}y =logy

## Natural (Naperian) logarithms The base is e.

Remember e is the irrational number where e = 2.71828... The symbol "ln" refers to natural logarithms.

log _{e}x =lnxln x is the exponent to which e must be raised to get x.

Why do we want to use logarithms? To simplify calculations in many cases.

Rules for logarithms

Product rule_{}

Quotient rule_{}

Power rule_{}This rule is useful because it allows us to solve equations where the variable is an exponent.

Exponential and Logarithmic Functions are inverse functionsConsider the following tables and the associated graphs: