Problem 36 - Algebra and Trigonometry

Question

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{11} 11 $$

Step-by-Step Solution

Answer

1

Step 1: Applying Logarithm Rule

The logarithm base \(b\) of a number \(x\) is the exponent to which \(b\) must be raised to yield \(x\). Thus, we want to find an exponent which when 11 is raised to, will result in 11.

Step 2: Calculate Logarithm

Since 11 raised to the power of 1 yields 11, we can conclude that \(\log _{11} 11 = 1\)

Key Concepts

Logarithm Rules

Understanding logarithm rules is fundamental when solving logarithmic questions. The logarithm rules simplify how we deal with logs and make calculations manageable.
Logarithms are, in essence, the inverse operations of exponents. For instance, the expression \( \log_b(x) \) asks the question: "To what power must the base \( b \) be raised, to obtain \( x \)?"

Here's a quick rundown of some essential rules:

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• Quotient Rule: \( \log_b \left( \frac{m}{n} \right) = \log_b(m) - \log_b(n) \)
• Power Rule: \( \log_b(m^n) = n \cdot \log_b(m) \)
• Base Base Rule: When the base and the argument are identical, \(\log_b(b) = 1\).

The base base rule plays a pivotal role, as seen in this problem: \( \log_{11}(11) = 1 \). This tells us 11 must be raised to the power of 1 to result in 11.

Exponents

Exponents are numbers that tell us how many times to multiply a base by itself. They act as shortcuts in math calculations by condensing repeated multiplication into a neat expression. For example, \(2^3\) means multiplying 2 three times: \(2 \times 2 \times 2\).

When dealing with logarithms, knowing basic exponent rules is advantageous. Logarithms and exponents are directly related because a logarithm represents the exponent's position to a certain number.

• Basic Rules: \(b^0 = 1\) for any base \(b\), which is foundational.
• \(b^1 = b\) indicates that any base raised to the power of 1 remains the base itself.
• If \(b^x = a\), then to reverse the operation, we express it as a logarithm: \(\log_b(a) = x\).

With our specific problem, raising 11 to the power of 1 yielded 11: \(11^1 = 11\), underlining the connection between exponents and logarithms.

Base of a Logarithm

In logarithms, the base is the number you're raising to an exponent. It's pivotal because it dictates the 'language' or scale used in the logarithmic operation.
For any logarithm, the base is what the logarithm "translates" to a power. When working with \(\log_b(x)\), think of \(b\) as the primary actor that lets you express \(x\) in terms of powers.

A few core points about bases:

• Often, the base is 10, termed "common logarithms," or \(e\) termed "natural logarithms." In these cases, the expressions simplify further.
• A base identical to the number in question simplifies to 1: meaning \(\log_{11}(11) = 1\).
• Understanding base relevance simplifies not just theoretical calculations but practical applications in science and engineering.

The base in our exercise was 11, and since the argument was also 11, it underscored the base base rule, making our solution straightforward.