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1. In N and a (¹1) are any two positive real numbers and for some real x, a^{x} = N, then x is said to be the logarithm of N to the base 'a'.
2. a^{x} = N is an exponential function and the corresponding logarithmic function is x = log_{a}N. 3. a^{logaN} = N. 4. The logarithm of the same number to different bases are different. 5. The logarithm of unity to any any non zero base is zero. 6. The logarithm of any nonzero positive number to the same base is unity. 7. log_{x}ab = log_{x}a + log_{x}b. 8. log_{x}(a/b) = log_{x}a  log_{x}b. 9. log_{x}a = log_{y}a + log_{x}y [This is known as change of base of a log]. 10. log_{x}y = log_{x}a / log_{y}a or log_{x}y = log_{a}y / log_{a}x. 11. log_{xn}y^{m} = m/n log _{x}y. 12. log a^{p}.b^{q} = p log a + q log b. 13. Logarithm of number contain, integral and decimal parts, the integral part of the logarithm of a number is called the "characteristic" and the decimal part is "mantissa". 14. If the number has 'n' digits the characteristic of its logarithm is 'n1' digits. 15. If the characteristic of the logarithm is n, the number will have 'n+1' digits.
Formulas1. If N = a^{x}, then log_{a}N = x. 2. Log _{a} 1 = log_{x} 1 = log_{5} 1 = 0. 3. Log _{a}a = 1. 4. Log MN = log M + log N. 5. log(M/N) = log M  log N. 6. log x^{n} = n log x.
7. log _{a} N = log_{b}N  log _{a}b. 
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