5.3: logarithms and their properties

power rule

the logarithm of a number raised by a power is equal to the exponent multiplied by the logarithm of the number. log(a)xˆr = rlog(a)x

product rule

the logarithm of the product of two numbers is equal to the sum of logarithms of the numbers. log(a)xy = log(a)x + log(a)y

quotient rule

the logarithm of the quotient of two numbers is equal to the difference of logarithms of the numbers. log(a)x/y = log(a)x - log(a)y

properties of logarithms

for a > 0, and a doesn't equal 1, and any real number k, the following hold. 1. log(a)1 = 0 2. log(a)aˆk = k 3. aˆlog(a)k = k, k > 0 product rule, quotient rule, and power rule.

logarithms

for all positive numbers a, where a doesn't equal 1, aˆy = x is equivalent to y = log(a)x. the expression log(a)x represents the exponent to which the base a must be raised in order to obtain x.

natural logarithm

for all positive numbers x, lnx = log(e)x

common logarithm

for all positive numbers x, logx = log(10)x

change of base theorem

for any positive numbers x, a, and b, where a doesn't equal 1 and b doesn't equal 1, log(a)x = log(b)x/log(b)a