Logarithm of a Product

This is the logarithmic form of the exponent law \((c^m)(c^n) = c^{m+n}\).

Proof:

Let \(\log_{c}(x) = m\) and \(\log_{c}(y) = n\).

Then, \(c^m = x\) and \(c^n = y\).

Now,

\(\therefore \log_{c}(xy)\) \(= \log_{c}(x) + \log_{c}(y)\), as required.

Logarithm of a Quotient

This is the logarithmic form of the exponent law \(\dfrac{c^m}{c^n} = c^{m-n}\).

Proof:

Let \(\log_{c}(x)= m\) and \(\log_{c}(y) = n\).

Rewriting each in exponential form gives \(c^m = x\) and \(c^n = y\).

Now,

\(\therefore \log_{c}\left(\dfrac{x}{y}\right)\) \(= \log_{c}(x) - \log_{c}(y)\), as required.

Logarithm of a Power

This is the logarithmic form of the exponent law \((c^m)^n = c^{mn}\).

Proof:

Let \(\log_{c}(x) = m\). Then, \(c^m = x\).

Now,

\(\therefore \log_{c}(x^n)\) \(= n\log_{c}(x)\), as required.