Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms. $$ \log _{10}(8.57)^{7} $$

Step-by-Step Solution

Answer

The value of \( \log_{10}(8.57)^7 \) is approximately 6.5275.

Step 1: Apply Logarithm Power Rule

The logarithm power rule states that \( \log_b(x^n) = n \cdot \log_b(x) \). In this case, \( \log_{10}(8.57)^7 = 7 \cdot \log_{10}(8.57) \).

Step 2: Convert Base 10 Logarithm to Natural Logarithm

Use the formula \( \log_a(x) = \frac{\ln(x)}{\ln(a)} \) to convert the base 10 logarithm to a natural logarithm. Here, it becomes \( 7 \cdot \log_{10}(8.57) = 7 \cdot \frac{\ln(8.57)}{\ln(10)} \).

Step 3: Calculate the Natural Logarithm

Use a calculator to find \( \ln(8.57) \approx 2.1471 \).

Step 4: Calculate the Natural Logarithm of 10

Find \( \ln(10) \approx 2.3026 \) using a calculator.

Step 5: Evaluate the Expression

Now compute the expression: \( 7 \cdot \frac{2.1471}{2.3026} \approx 7 \cdot 0.9325 \approx 6.5275 \).

Key Concepts

Logarithm Power Rule

Understanding the Logarithm Power Rule is essential when working with exponential terms inside logarithms. Imagine we have a number raised to a power, such as \( (8.57)^7 \). The rule comes in handy by allowing us to simplify the expression: crucially, \( \log_b(x^n) = n \cdot \log_b(x) \).

This rule states that the exponent can be brought in front of the logarithm as a multiplier.

It's like unraveling the power from the inside, making the expression simpler to solve.

For example, given our expression \( \log_{10}(8.57)^7 \), it can be rewritten as \( 7 \cdot \log_{10}(8.57) \).So, the problem becomes a straightforward multiplication, making further calculations more accessible.

Natural Logarithm

A natural logarithm is a logarithm with the base \( e \), where \( e \approx 2.71828 \). It's a common base in many mathematical contexts due to its unique properties, particularly in calculus and exponential growth models.

Symbolically represented as \( \ln(x) \).

Used frequently in continuous growth and decay processes.

The inverse operation of exponentiation with base \( e \).

In this particular problem, we use the conversion of a base 10 logarithm into a natural logarithm format when employing the formula \( \log_a(x) = \frac{\ln(x)}{\ln(a)} \).It allows for uniformity and often simplifies complex expressions. Here, using \( \ln(8.57) \) alongside \( \ln(10) \), we switch the base to help evaluate the number in a coherent manner.

Base Conversion

Base conversion in logarithms is crucial for evaluating expressions when our preferred calculator may not directly support certain bases. The conversion formula\( \log_a(x) = \frac{\ln(x)}{\ln(a)} \) allows us to change a logarithm to be computed in terms of natural logarithms.Here's how it helps:

Facilitates the computation using widely available \( \ln \) functions on calculators.

Converts the logarithm of any base to base \( e \), making the solution process consistent.

Applying the base conversion to the expression\( 7 \cdot \log_{10}(8.57) \), we reframe it as \( 7 \cdot \frac{\ln(8.57)}{\ln(10)} \).This transformation allows us to utilize natural logarithm calculations, conveniently facilitating our computations.

Mathematical Problem-Solving

Mathematical problem-solving isn't just about calculations; it's about applying the correct rules and formulas to find solutions efficiently. In the exercise given, problem-solving comes into play through several logical steps:

First, simplify the expression using logarithmic properties, specifically the power rule.

Next, use base conversion to move into a solvable format with tools readily available.

Then, perform the calculations step-by-step to ensure accuracy.

By following a systematic approach, complex problems can be broken down into manageable parts. Key aspects like verifying each step ensure the solution path remains error-free. In this exercise, successfully navigating from the power rule's simplification to evaluating the fully transformed logarithmic expression showcases how organized thinking and methodical action yield correct results.