Laws of Logarithms

Laws of Logarithms

Law 1

\[{\log _a}xy = {\log _a}x + {\log _a}y\]
Example

\(\begin{array}{l} {\log _5}25x\\ = lo{g_5}25 + {\log _5}x \end{array}\)

Beware!!
\[{\log _a}x + {\log _a}y \ne {\log _a}(x + y)\]

Law 2

\[{\log _a}\frac{x}{y} = {\log _a}x - {\log _a}y\]
Example
\(\begin{array}{l} {\log _5}\frac{x}{{25}}\\ = {\log _5}x + {\log _5}25 \end{array}\)

Beware!!
\[o{g_a}\frac{x}{y} \ne \frac{{{{\log }_a}x}}{{{{\log }_a}y}}\]

Law 3

\[{\log _a}{x^m} = m{\log _a}x\]
Example
\(\begin{array}{l} {\log _5}{y^5}\\ = 5{\log _5}y \end{array}\)


Beware!!
\[{({\log _a}x)^2} \ne 2{\log _a}x\]