Logarithms


It is a passive real number other than a^m = n
We write m=log_a x

10^3 =10004 then we can write log_{10} 1000 = 3
2^{-3} =\frac{1}{8} then we can write log_2 \frac{1}{8} = -3

Properties of logarithms

log_a (xy) = log_a x +log_a y
log_a (\frac{x}{y}) = log_a x - log_a y
log_x x = 1
Example 2^1=2 so we can write log_2 2 = 1
log_a 1 = 0
Example 5^0=1 in logarithm form log_5 1 = 0

log_a x^p = p(log_a x)
log_a x  = 1/log_x^a
log_a x = \frac{log_b^x}{log_b^a}   = \frac{log_⁡x}{log_⁡a}

Logarithms to the base 10 are known as common logarithms. When base is not mentioned it is taken as 10.
Characteristic – when the number is greater than 1 – the characteristic is one less from the number of digit in the left of the decimal point in the given number.

(\overline{48}) ̅.48= 1 (\overline{6185}) ̅.41 = 3
When the number is less than 1 one more than the number of zero between the decimal point & the 1st significant digit of the number & it is negative.
0.518 = -1 0.0347 = etc.

Log table-
10^{.001} = 10^{1/1000}   = 1.002305.238
10^{0.02} = 1.584293192
10^{0.01} = 10^{1/100}    = 1.023292992
10^{0.6} =3.981071706
10^{0.5} = 10^{\frac{1}{2}}= 3.16227766
10^{0.3} =  1.995262215
10^{0.1} = 10^{\frac{1}{10} =  1.258925412
10^{0.7} =  5.0118872336