Problem 17 - Precalculus with Limits

Question

Evaluate the function at the indicated value of \(x\) without using a calculator. \(f(x)=\log _{8} x \quad x=1\)

Step-by-Step Solution

Answer

The evaluated function at \(x=1\) is 0.

Step 1: Understanding the task

The task is to evaluate the function \(f(x) =\log _{8} x\) at \(x=1\). This is essentially finding the value of \(\log _{8} 1\).

Step 2: Using the properties of logarithms

According to the properties of logarithms, the log of 1 to any base is 0 because any number to the power of 0 equals 1.

Step 3: Calculating the value

So, with \(x=1\), the function \(f(x) =\log _{8} x\) equals \(\log _{8} 1\), which is 0.

Key Concepts

Properties of Logarithms

Logarithms have several important properties that make it easier to work with them in mathematical computations. One key property is that the logarithm of 1, regardless of the base, is always 0. This is because any number raised to the power of 0 results in 1. For example:

• \( \log_b 1 = 0 \) for any base \( b \)

Another essential property is the product rule, which states that the logarithm of a product is the sum of the logarithms:

• \( \log_b (xy) = \log_b x + \log_b y \)

There is also the quotient rule, which helps when dividing values:

• \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \)

The power rule states that the logarithm of a power is the exponent times the logarithm of the base:

• \( \log_b (x^k) = k \cdot \log_b x \)

Understanding these properties underpins the ability to simplify and calculate logarithmic expressions with confidence.

Base of Logarithms

The base of a logarithm is an integral component that indicates the number we use to repeatedly multiply itself to achieve another number. In our exercise, the base is 8, as indicated in the notation \( \log_8 x \). Various standard bases are often used in different contexts:

• **Common Logarithm:** The base 10 logarithms, often written as \( \log x \) or \( \log_{10} x \), are typically used in scientific calculations.
• **Natural Logarithm:** The base \( e \approx 2.718 \), known as the natural logarithm, is frequently encountered in calculus and exponential growth contexts and is denoted \( \ln x \).
• **Binary Logarithm:** The base 2 logarithm, sometimes written as \( \log_2 x \), is commonly used in computer science.

The base determines the rate of logarithmic growth and is fundamental in converting logarithms from one base to another. For instance, the change of base formula is given by:

• \( \log_b x = \frac{\log_k x}{\log_k b} \) for any base \( k \)

This aspect of logarithms is essential for understanding how logarithmic scales function across various applications.

Evaluating Logarithms

Evaluating logarithms means finding the value of a logarithm based on its base and the number it is applied to. In the given exercise, we need to evaluate \( f(x) = \log_8 x \) at \( x = 1 \). Recognizing that any number raised to the power of 0 equals 1 shows that \( \log_8 1 = 0 \).Here are some other simple evaluations of logarithms:

• \( \log_{10} 100 = 2 \), since \( 10^2 = 100 \)
• \( \log_2 8 = 3 \), because \( 2^3 = 8 \)
• \( \ln e = 1 \), since \( e^1 = e \)

To efficiently evaluate logarithms, one must understand the relationship between exponents and logarithms. Logarithms are essentially the opposite operation of exponentiation. The logarithmic expression \( \log_b x \) answers the question: "To what power must the base \( b \) be raised, to result in \( x \)?" This understanding is essential for solving logarithmic problems, as it highlights the inverse relationship they have with powers.