Problem 2 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{8}(13 \cdot 7) $$

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \( \log_8(13) + \log_8(7) \).

Step 1: Recall the Logarithmic Property

First, remember the product property of logarithms, which states that the logarithm of a product of numbers equals the sum of the logarithms of those numbers (i.e. \(log_b(M*N) = log_b(M) + log_b(N)\). This property allows us to separate the terms within the logarithm.

Step 2: Apply the Logarithmic Property

Applying this property to the expression \(\log_8(13*7)\), the logarithm of the product \(13*7\) is the sum of the logarithms of \(13\) and \(7\). Thus, \(\log_8(13*7)\) can be rewritten as \(\log_8(13) + \log_8(7)\).

Step 3: Evaluate the Logarithmic Expressions

Logarithms without a corresponding power of its base cannot be evaluated without a calculator. So, \(\log_8(13)\) and \(\log_8(7)\) will remain same as they cannot be simplified further without a calculator.

Key Concepts

Properties of Logarithms

Logarithms are powerful tools in mathematics, allowing us to perform complex calculations with ease. One set of rules that makes working with logarithms straightforward is the "properties of logarithms." These properties include the product property, quotient property, and power property.

Understanding these properties helps to greatly simplify logarithmic expressions and solve various mathematical problems efficiently.

• **Product Property:** The logarithm of a product equals the sum of the logarithms.
• **Quotient Property:** The logarithm of a quotient equals the difference of the logarithms.
• **Power Property:** The logarithm of a value raised to a power equals the exponent times the logarithm of the base value.

In essence, these properties enable the expansion of logarithmic expressions, making them easier to handle.

Product Property of Logarithms

The product property of logarithms is one of the key tools you can use to break down complex expressions. Formally, the property can be expressed as \[ \log_b(M \cdot N) = \log_b(M) + \log_b(N) \] This property allows us to separate items in a product into distinct, easier-to-manage terms.

When you have two numbers multiplied together inside a logarithm, this property tells you that it is equal to the sum of the two individual logarithms. This makes the calculations or further manipulations of expressions much more convenient.

Logarithmic Expressions

Logarithmic expressions are equations that involve logarithms, and these expressions can appear complex at first glance. However, by using properties of logarithms, these expressions can often be expanded and simplified.

In the original exercise example, the logarithmic expression \(\log_8(13 \cdot 7)\) is simplified using the product property. Here, the product within the logarithm can be separated into two simpler expressions: \(\log_8(13)\) and \(\log_8(7)\). This makes our computations easier or prepares the expression for further manipulation.

Calculator-Free Evaluation

One of the strengths of mastering the properties of logarithms is the capability for calculator-free evaluation. While some logarithms cannot be computed exactly without a calculator due to their complexity, being familiar with standard logarithmic values for base 10, base 2, and sometimes base \(e\) can be very helpful.

For instance,

• the fact that \(\log_{10}(10) = 1\),
• or that \(\log_{2}(8) = 3\) because \(2^3 = 8\).

In our exercise example, the logarithms \(\log_8(13)\) and \(\log_8(7)\) don't evaluate nicely without a calculator, but the knowledge that they can be expressed separately is useful both for mental math and algebraic manipulation.