We will use the exponent rule for products, \(a^p \cdot a^q = a^{p+q}\text{,}\) to prove that \(\log_a (xy) = \log_a x + \log_a y\) where \(a\text{,}\)\(x\) and \(y\) are greater than 0 and \(a \neq 1\text{.}\)
We will use the quotient rule of exponents, \(\frac{a^p}{a^q} = a^{p-q}\text{,}\) to prove that \(\log_a (\frac{x}{y}) = \log_a x - \log_a y\) where \(a\text{,}\)\(x\) and \(y\) are greater than 0 and \(a \neq 1\text{.}\)
We will use the power rule of exponents, \((a^p)^q=a^{pq}\text{,}\) to prove that \(\log_a x^k = k \log_a x\) where \(a\) and \(x\) are greater than 0, \(a \neq 1\) and \(q\) is a real number.
