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High School Mathematics
3.9 Important Points in Logarithms

1. In N and a (1) are any two positive real numbers and for some real x, ax = N, then x is said to be the logarithm of N to the base 'a'.

2. ax = N is an exponential function and the corresponding logarithmic function is x = logaN.

3. alogaN = 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 non-zero positive number to the same base is unity.

7. logxab = logxa + logxb.

8. logx(a/b) = logxa - logxb.

9. logxa = logya + logxy [This is known as change of base of a log].

10. logxy = logxa / logya or logxy = logay / logax.

11. logxnym = m/n log xy.

12. log ap.bq = 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 'n-1' digits.

15. If the characteristic of the logarithm is n, the number will have 'n+1' digits.

Formulas

1. If N = ax, then logaN = x.

2. Log a 1 = logx 1 = log5 1 = 0.

3. Log aa = 1.

4. Log MN = log M + log N.

5. log(M/N) = log M - log N.

6. log xn = n log x.

7. log a N = logbN - log ab.

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