Properties of Logarithms

Logarithms have specific properties that allow us to simplify complex equations. These rules—specifically the product, quotient, and power rules—let you take a single complicated logarithm and expand it into several simpler ones, or take several logarithms and condense them into one.

The Core Rules of Logarithms

For any base $b > 0$ (where $b \neq 1$) and positive numbers $x$ and $y$:

1. Product Rule: The logarithm of a product is the sum of the logarithms. $\log_b(xy) = \log_b x + \log_b y$

2. Quotient Rule: The logarithm of a quotient is the difference of the logarithms. $\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$

3. Power Rule: The logarithm of a number raised to an exponent is the exponent multiplied by the logarithm. $\log_b(x^n) = n \log_b x$

Change-of-Base Formula

Calculators usually only evaluate base 10 (common log) or base $e$ (natural log, $\ln$). The change-of-base formula lets you rewrite a logarithm in any base $c$:

$\log_b x = \frac{\log_c x}{\log_c b}$

For example, if you need to calculate $\log_3 7$, you can use the natural log: $\log_3 7 = \frac{\ln 7}{\ln 3}$.

Expanding Logarithmic Expressions

To expand an expression, apply the rules to break the logarithm apart as much as possible.

Example: Expand $\log_3\left(\frac{x^2 y}{z}\right)$

1. First, apply the Quotient Rule to separate the numerator and denominator: $\log_3(x^2 y) - \log_3(z)$
2. Next, apply the Product Rule to split the multiplied terms $x^2$ and $y$: $\log_3(x^2) + \log_3(y) - \log_3(z)$
3. Finally, use the Power Rule to move the exponent $2$ to the front: $2\log_3(x) + \log_3(y) - \log_3(z)$

Condensing Logarithmic Expressions

Condensing is the exact reverse of expanding. You want to combine multiple logarithms into a single logarithmic expression.

Example: Condense $2\ln x - \ln y + \frac{1}{2}\ln z$ into a single logarithm

1. First, apply the Power Rule to move the coefficients back into the logarithms as exponents: $\ln(x^2) - \ln(y) + \ln(z^{1/2})$
2. Notice that $\ln(x^2)$ and $\ln(z^{1/2})$ are positive, meaning their arguments belong in the numerator, while $-\ln(y)$ is negative, meaning $y$ belongs in the denominator. Apply the Product and Quotient Rules: $\ln\left(\frac{x^2 z^{1/2}}{y}\right)$ (Note: $z^{1/2}$ can also be written as $\sqrt{z}$, giving $\ln\left(\frac{x^2 \sqrt{z}}{y}\right)$.)