Laws of Logarithm –
1) logₐ (mn) = logₐ m + logₐ n (Product Law)
m
2) logₐ (------) = logₐ m – logₐ n (Quotient Law)
n
3) logₐ (mⁿ) = n (logₐ m) (Power Law)
4) logₐ a = 1
5) logₐ 1 = 0
6) logₐ aˣ = x
As, logₐ aˣ = x,
=> a = x.1 = x
a⁴ X b²
Example.) If, y = ----------, then log y = 4 log a + 2 log b – 3 log c
c³
Common Logarithms –
The logarithm to base 10 is called common logarithm. We denote log₁₀ m by log m only.
Example: i) (10)¹ = 10
=> log₁₀ 10 = 1
=> log 10 = 1
=> log (10)¹ = 1 (Ans.)
Example: ii) (10)² = 100
=> log₁₀ 100 = 2
=> log 100 = 2
=> log (10)² = 2
Thus, log (10)¹ = 1, log (10)² = 2, etc (Ans.)
Example: iii) (10)ˉ¹ = - 1,
=> log₁₀ (10)ˉ¹ = - 1
=> log 10 = - 1
=> log (10)ˉ¹ = - 1 (Ans.)
Example: iv) (10)ˉ² = - 2
=> log₁₀ (10)ˉ² = - 2
=> log 10 = - 2
=> log (10)ˉ² = - 2 (Ans.)
If no base is given, the base is always taken as 10,
as a example log 5 = log₁₀ 5, log 7 = log₁₀ 7