91Ó°ÊÓ

Understanding the properties of logarithms is crucial for manipulating and simplifying logarithmic expressions effectively. These properties serve as tools that help combine, expand, or condense logarithmic expressions. Here are some primary properties you should remember:

• **Product Property**: This property states that the logarithm of a product is the sum of the logarithms. In mathematical terms, \( \log_b(M \cdot N) = \log_b(M) + \log_b(N) \). This means that if you're adding two logs with the same base, you can combine them by converting into the log of a product.


• **Quotient Property**: The logarithm of a quotient represents the difference of the logarithms. This is expressed as \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \). This property is useful for simplifying fractions involving logarithms.


• **Power Property**: The logarithm of a power means that you can bring the exponent in front of the log. The formula is \( \log_b(M^k) = k \cdot \log_b(M) \). This property helps to simplify expressions where a term is raised to a power inside a logarithm.

Using these properties, you can condense or expand logarithmic expressions, making calculations and solving equations manageable.

Condensing logarithmic expressions involves using the properties of logarithms to merge multiple log terms into a single log. This is the opposite of expanding logs and is a powerful way to simplify complex expressions.

In the given exercise, we condensed the expression \( \log 5 + \log 2 \) into a single log, \( \log(5 \cdot 2) \), thanks to the product property. When two or more logs with the same base are added together, they can be combined by multiplying their contents inside a single logarithm.

This skill is especially helpful when solving equations or simplifying expressions, as it reduces the number of terms you have to deal with. It also often leads you directly to a solution, like in our example where ultimately \( \log(10) \) simplifies easily to the known value of 1.

Make sure to simplify expressions completely. When you encounter sum or difference of logs, always look for opportunities to combine terms using these properties.

The base 10 logarithm, often written as \( \log \) without a base, is one of the most commonly used logarithms in mathematics and science. It is also known as the common logarithm. The concept of base 10 logging is rooted in powers of ten, which is a foundation of our decimal system.

Understanding base 10 logarithms can be straightforward with key facts:

• \( \log_{10}(10) = 1 \) because \( 10^1 = 10 \).


• \( \log_{10}(100) = 2 \) because \( 10^2 = 100 \).


• \( \log_{10}(1) = 0 \) since \( 10^0 = 1 \).

In the exercise example, when we combined \( \log 5 + \log 2 \) and simplified it to \( \log 10 \), we used the knowledge that \( \log_{10}(10) \) equals 1 to finish simplifying the expression.

This simplification is due to the identity that relates logs and exponents directly back to simple integer values. Mastering the base 10 logarithm is quite useful because it streamlines evaluating and simplifying expressions in many contexts, from basic arithmetic to more complex scientific calculations.