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(1)First Law (Product Law) :

The logarithm of a product is equal to the sum of the logarithms of the factors,
i.e. , log_a(m x n) = log_am + log_an.

Proof :Let log_a m = x and log_a n = y. Then,
m = a^x and n = a^y.
m x n = a^x x a^y = a^x+y .
Changing in logarithmic form, we have
log_a(mn)= x + y
Substituting the values of x and y, we get
log_a(mn) = log_am + log_an.
The above result is capable of extension i.e.,
log_a(mnp....)=log_am + log_an + log_ap + .....

Remember : log_a(m + n) is not equal to log_am + log_an, in geneeral.

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(2) Second Law (Quotient Law);

The logarithm of a fraction is equal to a the difference between the logarithm of numerator and the logarithm of denominator, i.e.,
log_a m/n = log_am - log_an.

Proof :log_am = x and log_an = y. Then,
a^x = m and a^y = n
m/n = a^x/a^y = a^x-y
Changing in logarithmic form , we have
log_a m/n = x - y
Substituting the values of x and y, we get
log_a m/n = log_am - log_an.

Remember : log_am/log_an is not equal to log_am - log_an.

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(3) Third Law (Power Law) :

The logarithm of a power of a number is equal to the logarithm of the number multiplied by the power,i.e,
log_a(m)ⁿ = n log_a m.

Proof : log_am = x
Then , a^x = m
mⁿ =( a^x)ⁿ = a^xn = a^nx
Changing in logarithmic form, we have
log_amⁿ = nx
Substituting the value of x, we get
log_amⁿ = n log_a m.

Remember : log_amⁿ = n log_a m

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(4) Rule of change of base :

(i) log_a m =[(log_b m/log_b a)]

Proof : Let log_a m = x, a^x = m
Taking log to the base b, both sides,
log_b (a^x) = log_b m
x log_b a = log_b m
Now putting the value x = log_a m
(log_a m) log_b a = log_bm

Remark :If b = m then
log_am = log_mm/log_ma = 1/log_ma

(ii) Prove log_ba x log_ab = 1.

Proof : Let = log_ba x log_ab
log_ma/log_mb x log_mb/log_ma = 1
log_ba x log_ab = 1

(iii) Prove log_bm = - log_1/b(m)

Proof : Let = log_bm = p
b^p = m
(1/b)^-p = m
log_1/bm = -p = -log_bm
log_bm = -log_1/bm.

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