Problem 3 - Calculus of a Single Variable

Question

Evaluating a Logarithmic Expression In Exercises \(1-4\) , evaluate the expression without using a calculator. $$ \log _{7} 1 $$

Step-by-Step Solution

Answer

The evaluation of the expression \( \log_{7} 1 \) is 0.

Step 1: Understand the problem

The problem presents a logarithm \( \log_{7} 1 \) and asks for it to be solved without using a calculator. In a logarithm \( \log_{b} n = x \), 'b' is the base, 'n' is the result and 'x' is the exponent. In other words, it is asking: to what power (x) should we raise the base (b) to get the result (n). In our case, base is 7, and result is 1.

Step 2: Apply the properties of a logarithm

Any number (except for 0) raised to the power of 0 equals 1. Thus, \( b^0 = 1 \). Hence, in our case, 7 to the power of what equals to 1? The answer is clearly 0 as any non-zero number raised to 0 is 1.

Step 3: Write the answer

So, we can write \( \log_{7} 1 = 0 \).

Key Concepts

Logarithm Properties

Logarithms are powerful mathematical tools that can simplify complex calculations and help solve for unknowns in exponential expressions. One important property of logarithms that's particularly useful when evaluating them is the fact that any base raised to the exponent of zero equals one, represented as \( b^0 = 1 \).

Moreover, logarithms have unique characteristics that make them versatile in problem-solving. Key properties include:

• The Product Property: \( \log_b (mn) = \log_b m + \log_b n \), useful for breaking down multiplication into addition.
• The Quotient Property: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \), which simplifies division to subtraction.
• The Power Property: \( \log_b (m^n) = n \cdot \log_b m \), helpful in dealing with exponents within logarithms.

Understanding these properties is critical, as they allow for flexibility when manipulating and evaluating logarithmic expressions.

In the problem, the use of the property \( b^0 = 1 \) explains why \( \log_7 1 = 0 \). All the properties together form a toolkit for tackling various logarithmic challenges.

Base of a Logarithm

The base of a logarithm is crucial as it dictates the scale of the logarithmic function. It’s the number that is raised to a certain power to yield another number. In the expression \( \log_b n = x \), 'b' is the base. For the exercise given, the base is 7.

Understanding the base is important because:

• It defines what power you're solving for. For example, \( \log_7 1 \) means you're looking for what power 7 must be raised to result in 1.
• Different bases yield different results for the same logarithmic expression, making it important to never overlook the base.
• Common bases like 10 (common logarithm), e (natural logarithm), or 2 (binary logarithm) have specific applications in science, engineering, and computing.

Intuitively grasping the concept of a base allows you to better understand the behavior of logarithmic functions and their applications in various scientific fields.

Exponentiation Concepts

Exponentiation is the process of raising a number to a power, symbolized as \( b^x \), where 'b' is the base and 'x' is the exponent. It’s a core concept in understanding logarithms.

Key points about exponentiation include:

• Exponents indicate how many times to multiply the base by itself. For example, \( 7^2 \) means 7 multiplied by itself, i.e., 49.
• When the exponent is 0, any non-zero number raised to this power is 1. That's why \( 7^0 = 1 \), which was used to solve the original problem: \( \log_7 1 = 0 \).
• Exponents can be fractions, which indicate roots, such as \( b^{1/2} \) representing the square root of 'b'.

This comprehensive understanding of exponentiation aids in demystifying logarithms, as logarithms are essentially the reverse operation of exponentiation. In the exercise, knowing \( 7^0 = 1 \) helped quickly solve the problem at hand.