Rules of logarithms

To manipulate logarithmic expressions and solve equations, we use several key rules.

Formula: Logarithmic rules

For positive $x, y, a$ and $a \neq 1$ :

Product Rule: $\log_{a} (x y) = \log_{a} x + \log_{a} y$

Quotient Rule: $\log_{a} (\frac{x}{y}) = \log_{a} x - \log_{a} y$

Power Rule: $\log_{a} (x^{n}) = n \log_{a} x$

Special values:

$\log_{a} a = 1$

$\log_{a} 1 = 0$

In particular, $\ln e = 1$ and $\ln 1 = 0.$

Example A: Simplifying log expressions

Question

Simplify the expression: $\log_{2} 3 + 2 \log_{2} 10 - \log_{2} 150.$

Solution

$\begin{matrix} \log_{2} 3 + 2 \log_{2} 10 - \log_{2} 150 & = \log_{2} 3 + \log_{2} (10^{2}) - \log_{2} 150 & (Power rule) \\ = \log_{2} 3 + \log_{2} 100 - \log_{2} 150 \\ = \log_{2} (3 \times 100) - \log_{2} 150 & (Product rule) \\ = \log_{2} 300 - \log_{2} 150 \\ = \log_{2} (\frac{300}{150}) & (Quotient rule) \\ = \log_{2} 2 \\ = 1 ∎ \end{matrix}$