### Quotient rule of Logarithms

log_a X - log_a Y = log_a (X/Y)

Subjects > Mathematics > Logarithms

#### The product rule of Logarithms

Provided that X > 0, Y > 0 and a ne 1 and n, m is any real number, then
log_a X - log_a Y = log_a (X/Y)

#### Proof

let X = a^n and Y = a^m
From the definition of logarithms, it is true that:
⇒ log_a X = n and log_a Y = m
Now,
log_a (X/Y) = log_a (a^n / a^m) - By substituting the values of X and Y
⇒ log_a (X/Y) = log_a a^(n - m) - Quotient rule of exponents
⇒ log_a (X/Y) = n - m
Now substitute the values of n and m
⇒ log_a (X/Y) = log_a X - log_a Y
∴ Hence proved.

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#### The product rule of Logarithms

Provided that X > 0, Y > 0 and a ne 1 and n, m is any real number, then
log_a X - log_a Y = log_a (X/Y)

#### Proof

let X = a^n and Y = a^m
From the definition of logarithms, it is true that:
⇒ log_a X = n and log_a Y = m
Now,
log_a (X/Y) = log_a (a^n / a^m) - By substituting the values of X and Y
⇒ log_a (X/Y) = log_a a^(n - m) - Quotient rule of exponents
⇒ log_a (X/Y) = n - m
Now substitute the values of n and m
⇒ log_a (X/Y) = log_a X - log_a Y
∴ Hence proved.

Recommended Lessons