Definition:-
We define log as, a^y = x than y = log_ax. in log_ax. both x and a are positive ie. x > 0 and a > 0 and also a 1.

Dumb Question:- Why a cannot be 1 ?
Ans:- Suppose a is 1 then let us attempt to y such that y = log₁x.
Now according to definition of log.
1^y = x.
But no matter what power we raise to 1 the answer will be. 1 only so we will never never be able to find y.
Hence a cannot be 1.

Some important formulae:- (Formulaes marked with * are important. This is not be printed)
1. log_aa = 1.
2. log_any1 = 0.
3. log_ca = log_ba.log_cb.
Why ?
Let log_ba = x and log_cb = y
So, by definition, a = b^x ..................................... (i)
b = c^y ...................................... (ii)
Using (i) & (ii) a = c^xy
Now taking log on both sides.
log_ca = xy
log_ba.log_cb.

Illustration - 1.
Find value of log₂10.log₁₀2 ?
Using formula 3 we get.
log₂10.log₁₀2 = log₁₀10
Now using formula 1 we get
log₁₀10 = 1
Hence log₂10.log₁₀2 = 1

4. log_a(m.n) = log_am + log_an
Why ?
Let log_am = x
log_an = y
So, m = a^x and n = a^y [Using definition]
m.n = a^x.a^y = a^{x + y}
=> log_a(mn) = log_aa^{x + y}
                   = x + y
                   = log_am + log_an.