Question

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms. $$ \log _{11}(8.12)^{1 / 5} $$

Short answer

The logarithm is approximately 0.1746.

Step-by-step solution

Simplify the Expression

Begin by using the power rule for logarithms, which states that \( \log_b(x^n) = n \log_b(x) \). Therefore, \( \log_{11}(8.12)^{1/5} = \frac{1}{5} \log_{11}(8.12) \).

Apply Change of Base Formula

Use the change of base formula \( \log_{a} x = \frac{\ln x}{\ln a} \) to convert the logarithm. Thus, \( \log_{11}(8.12) = \frac{\ln(8.12)}{\ln(11)} \).

Combine the Formulas

Substitute the expression from Step 2 into the expression from Step 1: \[ \frac{1}{5} \log_{11}(8.12) = \frac{1}{5} \times \frac{\ln(8.12)}{\ln(11)} = \frac{\ln(8.12)}{5 \ln(11)} \].

Calculate Using a Calculator

Plug the values into a calculator: compute \( \ln(8.12) \approx 2.093 \) and \( \ln(11) \approx 2.398 \). Then, \( \frac{2.093}{5 \times 2.398} \approx \frac{2.093}{11.99} \approx 0.1746 \).

Key concepts

Logarithm Properties

Logarithms have various properties that make them incredibly useful for solving complex mathematical equations. One important property to remember is the **power rule**, which states that \( \log_b(x^n) = n \log_b(x) \). This rule helps simplify expressions involving exponents. For example, if you need to find \( \log_{11}(8.12)^{1/5} \), you can first use the power rule to get \( \frac{1}{5} \log_{11}(8.12) \).
Another crucial property is the **change of base formula**. This formula allows you to convert a logarithm of any base into natural logarithms (\( \ln \)). This is especially useful when you do not have a specific base function on your calculator. The formula is \( \log_{a} x = \frac{\ln x}{\ln a} \).
With these properties, solving logarithmic expressions becomes manageable, even when they look tricky at first. They help us break down the problems step-by-step, making it easier to compute with tools like calculators.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). It's one of the most commonly used logarithms in mathematics because it arises naturally in calculus and many real-world applications.
**Why Use \( \ln x \)?**

• Natural logarithms simplify the differentiation and integration processes in calculus since derivatives and integrals of \( \ln x \) have elegant solutions.
• They appear in continuously compounded interest in finance, population growth models in epidemiology, and in the time-to-failure analysis in engineering.

When tackling a problem like \( \log_{11}(8.12) \), converting to a natural logarithm using the change of base formula makes it more straightforward to solve. You compute \( \ln(8.12) \approx 2.093 \) and \( \ln(11) \approx 2.398 \), and then perform simple arithmetic to solve the problem efficiently.

Calculus

Calculus, the study of how things change, often employs logarithms, especially natural logarithms. **Differentiation and Integration**, the core operations in calculus, frequently use logarithms. Here’s why calculus students are encouraged to understand logarithmic concepts:

• **Differentiation**: The derivative of \( \ln x \) is \( \frac{1}{x} \), making it simple to work with in calculus problems. It’s also a key part in solving growth and decay problems.
• **Integration**: Integrating a function involving \( \ln x \) can often be solved by recognizing patterns or using integration by parts. For example, integrating \( \frac{1}{x} \) leads directly to \( \ln x + C \, (C \text{ is the integration constant}) \).

Logarithms are common in calculus problems involving exponential growth or decay, optimization, and many other areas. Understanding their properties can simplify these complex questions, making calculus less intimidating and more approachable.