Problem 5 - Precalculus

Question

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\log _{10} 10^{8}$$

Step-by-Step Solution

Answer

The expression \( \log_{10} 10^8 \) simplifies to 8.

Step 1: Identify the Property to Apply

We begin by recognizing that the expression \( \log_{10} 10^8 \) involves a logarithm of a power. The primary property that applies here is the power rule of logarithms: \( \log_b (b^x) = x \). This property allows us to simplify the expression by focusing on the exponent.

Step 2: Apply the Logarithm Power Rule

According to the power rule, \( \log_{10} (10^8) \) simplifies directly to \( 8 \) because the base of the logarithm (10) matches the base of the exponent (10). Thus, the expression simplifies to \( 8 \) without performing any additional calculation.

Key Concepts

Properties of Logarithms

Logarithms have unique properties that make them a powerful tool for simplifying expressions involving exponents and multiplicative relationships. Understanding these properties helps to manipulate and solve logarithmic expressions without necessarily using a calculator. Some fundamental properties of logarithms include:

• Product Rule: This states that the logarithm of a product is the sum of the logarithms: \( \log_b (MN) = \log_b M + \log_b N \).
• Quotient Rule: This states that the logarithm of a quotient is the difference of the logarithms: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \).
• Power Rule: The focus of our current exercise, this rule states that the logarithm of a power is the exponent times the logarithm: \( \log_b (M^x) = x \log_b M \).

Each property provides a method to adjust complex expressions into more manageable forms, facilitating simplification or further analysis. The power rule is particularly advantageous when the base of the logarithm and the base of the exponent are the same, as in our exercise with \( \log_{10} (10^8) \).

Power Rule

The power rule of logarithms is pivotal for simplifying expressions where a logarithm involves an exponent directly. It simplifies the expressions by moving the exponent out front, as a multiplier to the logarithm. For the power rule:

• Formula: \( \log_b (b^x) = x \). This directly implies that whenever the base of the logarithm \( b \) matches the base of the exponential expression inside the logarithm, the logarithm simplifies to just the exponent \( x \).
• Example: Consider\(\log_{10} (10^8)\). According to the power rule, because the base \(10\) is the same for both the logarithm and the exponent, the expression simplifies directly to \( 8 \).

Since the power rule allows the logarithm and exponential operations to counteract each other, it essentially "lifts" the exponent out of the logarithmic expression, leading to quick solutions without additional computation.

Logarithm Simplification

Simplifying logarithms involves transforming a given expression into its simplest form using the properties of logarithms. This often requires combining, splitting, or adjusting exponentials and products within a logarithmic function.

Here's how you can apply simplification techniques, generally driven by recognizing patterns or properties:

• Identify the Pattern: Look for expressions involving multiplication, division, or powers over the logarithm that match the product, quotient, and power rules.
• Apply the Properties: Use the relevant logarithmic properties. For example, with an expression like \( \log_{10}(10^8) \), immediately recognize the power rule applies, leading instantly to simplification as \( 8 \).
• Check Consistency: Ensure the calculation is consistent with the properties. After simplifying, the result should logically follow from the original expression.

By understanding and employing the properties of logarithms, complex expressions can be made more manageable, just as in the given exercise where using the power rule led directly to the simplified answer.