Lesson 5: Properties of Logarithms

Property

Since logarithms are inverse functions of exponentials, logarithm properties can be derived from the corresponding exponential properties:

From $b^m \cdot b^n = b^{m+n}$, we derive $\log_b(xy) = \log_b(x) + \log_b(y)$

Property

The properties of logarithms can be used to rewrite expressions. You can expand a single logarithm into a sum or difference of simpler logs, or combine multiple logs into a single, more complex logarithm.

Expand: $\log_b(x^A y^B) = A \log_b x + B \log_b y$

Combine: $A \log_b x - B \log_b y = \log_b(\frac{x^A}{y^B})$

Property

To condense logarithmic expressions with the same base into one logarithm, we start by using the Power Property to get the coefficients of the log terms to be one and then the Product and Quotient Properties as needed.

Examples

• Condense $2\log_3 x + 5\log_3 y$. This becomes $\log_3 x^2 + \log_3 y^5 = \log_3(x^2 y^5)$.
• Condense $\ln 10 - 3\ln z$. This becomes $\ln 10 - \ln z^3 = \ln(\frac{10}{z^3})$.
• Condense $\log_2 40 - \log_2 5$. Using the quotient property, this becomes $\log_2(\frac{40}{5}) = \log_2 8 = 3$.

Explanation

Condensing is the reverse of expanding. You use the log properties backwards to combine a sum or difference of logs into a single, more compact logarithm. This is key for simplifying and solving log equations.