Problem 23 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to expand the expression. $$\log 6^{10}$$

Step-by-Step Solution

Answer

The expanded form is \( 10 \cdot \log(6) \).

Step 1: Apply logarithm properties

Use product, quotient, and power rules of logarithms.

Step 2: Simplify

The result is The expanded form is \( 10 \cdot \log(6) \)..

Key Concepts

Power Rule for Logarithms

The Power Rule for Logarithms is a crucial tool for simplifying logarithmic expressions that involve exponents. The rule states: \[ \log_b(a^n) = n \cdot \log_b(a) \]This formula means that when you have a logarithm of a number raised to a power, you can move that exponent to the front as a multiplier. This transformation is particularly useful when expanding expressions.
Think of it this way: if there is a small exponent trapped inside, the Power Rule lets us pull it out and multiply. The power becomes a coefficient in front of the log expression. By applying this rule, you make the expression easier to manage, especially when solving equations or analyzing logarithms.

Logarithmic Expressions

Logarithmic expressions are mathematical phrases involving logarithms, and they are used in various applications, such as solving exponential equations and analyzing scientific data. A logarithm tells us the power to which a base number must be raised to produce a given number.
Logarithms can use different bases, but the most common are base 10 (common logarithm) and base \(e\) (natural logarithm). Knowing how to manipulate these expressions using the laws of logarithms, like the Power Rule, allows computational flexibility and understanding of their behaviors.

• For example, \( \log_{10}(100) = 2 \) because 10 raised to the power of 2 equals 100.
• Natural logarithms are denoted as \( \ln \), where \( \ln(e^2) = 2 \). The base \( e \) is a mathematical constant approximately equal to 2.718.

Understanding logarithms helps simplify complex calculations by making multiplicative processes become additive, especially when employing the laws efficiently.

Expanding Logarithms

Expanding logarithms involves using the laws of logarithms to rewrite a single logarithm in terms of simpler, multiple logarithmic expressions. The process often employs rules such as the Power Rule, the Product Rule (\( \log_b(xy) = \log_b(x) + \log_b(y) \)), and the Quotient Rule (\( \log_b(x/y) = \log_b(x) - \log_b(y) \)).
When expanding expressions, the goal is to break down a compound log into more manageable parts. Let's look at an example:- For the expression \( \log(6^{10}) \), we can use the Power Rule: move the 10 outside to get \( 10 \cdot \log(6) \). This is an expanded form.- If there were multiplication or division inside the logarithm, you could apply other rules to further split the terms.Expanding logarithms is particularly useful in algebra and calculus, simplifying expressions for integration or derivation. It helps mathematicians and students see underlying patterns and relationships in complex equations.