Logarithms

Synopsis

 Related Terms

• Let a, b, c are three numbers and they are related so that a^b = c ; then exponent, 'b' is called the logarithm of number, 'c' to the base 'a', and  log_a c= b
• Definition of logarithm: Thus, logarithm of any number to a given base is equal to the index to which the base should be raised to get the given number.  

Important points

• The exponential form: a^b = c
• Logarithmic form: log_a c = b
• When x0 = 1 ⟹  log_x 1 = 0
• Logarithm of 1 to any base is zero. 
•  Since,a¹ =a,log_a a =1
• Logarithms to the base 10 are known as common logarithms.
• If no base is given, the base is always taken as 10.
• If a and x are positive real numbers, where a ≠ 1, then  a^log_ax =x
• For a > 0, a ≠ 1, log_a x =log_b y ⟹ x= y (x,y>0).
• If a > 1 and x > y, then  log_a x > log_a y
• If 0 < a < 1 and x > y, then  log_a x > log_a y

Laws of Logarithms
For m,n, a > 0 and a ≠ 1 

• Product Law: The logarithm of a product is equal to the sum of the logarithms of its factors.
  i.e. log_a (m × n) = log_a m + log_a n
  
  
• Quotient Law: The logarithm of fraction is equal to the difference between the logarithm of the numerator and the logarithm of the denominator.
  log_a  = log_a m – log_a n
• Power Law: The logarithm of a power of a number is equal to the logarithm of the number multiplied by the power.
  loga mⁿ = n log_a m

• Logarithm of a number x to the base a is equal to 1 divided by logarithm of a to the base x.
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• log_ax = log_b x log_a b
• x^{log_a y} = y^log_ax
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• If log_ab =x for all a > 0, a ≠ 1, b > 0 and x ∈ R, then log_1/a b = - x, log_a 1/b = - x and  log_1/a 1/b= x