Logarithms

Let a and N be positive real numbers and let N = aⁿ. Then n is called the logarithm of N to the base a. We write this as n = log_a N.

Find the value of log₃ 27 is

Rewrite as an equation:

log₃ 27 = x

Rewrite log₃ 27 = x in exponential form using the definition of a logarithm. If x and b are positive real numbers and b does not equal 1, then log_b x = y is equivalent to b^y=x

3^x=27 <=> 3^x= 3³ <=> x = 3

Rules of Logarithms:

The following important rules apply to logarithms.

Rule name	Rule
Logarithm product rule	logb(x ∙ y) = logb(x) + logb(y)
Logarithm quotient rule	logb(x / y) = logb(x) – logb(y)
Logarithm power rule	logb(x y) = y ∙ logb(x)
Logarithm base switch rule	logb(c) = 1 / logc(b)
Logarithm base change rule	logb(x) = logc(x) / logc(b)
Derivative of logarithm	f (x) = logb(x)⇒ f ‘ (x) = 1 / ( x ln(b) )
Integral of logarithm	∫logb(x) dx = x ∙ ( logb(x)- 1 / ln(b)) + C
Logarithm of negative number	logb(x) is undefined when x≤ 0
Logarithm of 0	logb(0) is undefined
Logarithm of 1	logb(1) = 0
Logarithm of the base	logb(b) = 1
Logarithm of infinity	lim logb(x) = ∞,when x→∞