Problem 18 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log M^{-8} $$

Step-by-Step Solution

Answer

The simplified form of the given expression according to the properties of logarithms is \( -8 \log M \).

Step 1: Recognize the Power Rule

The given expression can be rewritten using the power rule of logarithms, which allows the exponent to be moved in front of the logarithm.

Step 2: Apply the Power Rule

We have \( \log M^{-8} \). This can be rewritten as \( -8 \log M \), applying the power rule of logarithms.

Key Concepts

Power Rule

Understanding the power rule is essential when dealing with logarithms, especially for simplifying expressions. The power rule for logarithms states that if you have a logarithmic expression with an exponent, such as \( \log_b (M^n) \), you can bring the exponent \( n \) in front of the log to simplify it. This gives you \( n \cdot \log_b M \).

For example, in the expression \( \log M^{-8} \), the power rule allows us to move \(-8\), the exponent, to the front of the logarithm. This transforms \( \log M^{-8} \) into \(-8 \log M\), making it easier to work with. This rule is incredibly helpful when expanding or simply evaluating logarithmic expressions.

Logarithmic Expressions

Logarithmic expressions are equations that contain logarithms, which are operations that determine how many times a certain base number is multiplied to reach another number. They are the inverse operations of exponents.

For instance, the logarithmic expression \( \log_b x \) asks "To what power must we raise \( b \) to get \( x \)?". Simplifying these expressions often involves using the laws of logarithms to transform them into more manageable forms, particularly when more than one operation or exponent is involved.

In general, breaking down logarithmic expressions using rules such as the power rule, product rule, and quotient rule is vital for problem-solving.

Properties of Logarithms

Logarithms come with several key properties that help simplify complex expressions. Here are some important ones:

• Power Rule: This allows moving the exponent in front of the logarithm, simplifying the expression.
• Product Rule: \( \log_b (XY) = \log_b X + \log_b Y \). This property splits logarithms of a product into sums.
• Quotient Rule: \( \log_b \left(\frac{X}{Y}\right) = \log_b X - \log_b Y \). This deals with division inside the log.
• Change of Base Formula: Helps to convert between different logarithmic bases, especially useful when using calculators.

Understanding these properties is fundamental for manipulating and simplifying logarithmic expressions effectively.

Exponents

Exponents are a means of expressing repeated multiplication of a number by itself. For example, in \( x^n \), \( n \) is the exponent, and it indicates that \( x \) is multiplied by itself \( n \) times. Exponents are intimately connected to logarithms as they act as inverse operations.

For instance, for any expression \( a^x = b \), the corresponding logarithm would be \( \log_a b = x \). This relation shows how logarithms can be seen as the opposite of exponentiation, finding the power needed to reach a certain number using a given base.

Working with exponents within logarithmic expressions often requires employing logarithm properties, such as the power rule, to simplify and solve equations proficiently.