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# Logarithms

## Logarithms Synopsis

Synopsis

Related Terms

• Let a, b, c are three numbers and they are related so that ab = c ; then exponent, 'b' is called the logarithm of number, 'c' to the base 'a', and  loga 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: ab = c
• Logarithmic form: loga c = b
• When x0 = 1   logx 1 = 0
• Logarithm of 1 to any base is zero.
•  Since,a1 =a,loga 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  alogax =x
• For a > 0, a ≠ 1, loga x =logb y ⟹ x= y (x,y>0).
• If a > 1 and x > y, then  loga x > loga y
• If 0 < a < 1 and x > y, then  loga x > loga 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. loga (m × n) = loga m + loga 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 = loga m – loga 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 loga m
• Logarithm of a number x to the base a is equal to 1 divided by logarithm of a to the base x.
• logax = logb x loga b
• xloga y = ylogax
• If logab =x for all a > 0, a ≠ 1, b > 0 and x ∈ R, then log1/a b = - x, loga 1/b = - x and  log1/a 1/b= x