Problem 30 - Precalculus

Question

Evaluate the expression. \(\log _{13}(\sqrt{13})\)

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{1}{2}\).

Step 1: Understand the Expression

The given expression is \(\log_{13}(\sqrt{13})\). This is the logarithm of \(\sqrt{13}\) with base 13. The task is to evaluate this expression, meaning to find the value of this logarithm.

Step 2: Re-write the Square Root

Recognize that the square root of 13 can be written using fractional exponents. Therefore, \(\sqrt{13}\) is equivalent to \(13^{1/2}\). This allows us to work with the expression more conveniently in terms of powers.

Step 3: Apply Logarithm Power Rule

Use the logarithm power rule, which states that \(\log_b(a^c) = c\log_b(a)\). Apply this rule to the expression \(\log_{13}(13^{1/2})\), which becomes \(\frac{1}{2}\log_{13}(13)\).

Step 4: Simplify Logarithmic Expression

Recall that \(\log_b(b) = 1\) for any base \(b\). Therefore, \(\log_{13}(13) = 1\). Substitute this into the previous expression, yielding \(\frac{1}{2} \times 1 = \frac{1}{2}\).

Key Concepts

Fractional Exponents

Logarithms and fractional exponents go hand in hand, especially when simplifying expressions like square roots. A fractional exponent represents a power operation. This appears commonly in expressions that involve roots.
For example,

• The square root of a number can be expressed with a fractional exponent as \( x^{1/2} \), where the fraction indicates the root.
• A cube root would be written as \( x^{1/3} \).

When dealing with fractional exponents, it allows us to use the properties of exponents more flexibly in cooperation with logarithms. This means you can rewrite roots as powers, which is particularly helpful when applying the power rule in logarithms. Converting roots to exponents makes it easier to manipulate and simplify logarithmic expressions.

Power Rule

The power rule in logarithms is a powerful tool that helps to simplify expressions involving logarithms and exponents. This rule makes it possible to bring down exponents as coefficients, which can be very handy.
The power rule is stated as:

• \( \log_b(a^c) = c \times \log_b(a) \)

This means that when a number is raised to a power inside a logarithm, the exponent can be moved in front of the logarithm as a multiplier. In the exercise given, converting \( \sqrt{13} \) to \( 13^{1/2} \) allows us to use the power rule: \( \log_{13}(13^{1/2}) \) becomes \( \frac{1}{2}\log_{13}(13) \).
Ultimately, the power rule makes solving logarithmic functions much more straightforward by reducing complex expressions into simpler, more manageable pieces.

Logarithmic Identities

Understanding logarithmic identities is crucial in manipulating and simplifying logarithmic expressions. One of the most fundamental identities is:

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

This identity states that the logarithm of a number at its own base equals one. This concept is especially useful when simplifying expressions, as seen in the exercise \( \log_{13}(13) \) = 1.
Logarithmic identities are rules or properties that help simplify expressions and solve logarithmic equations. Given that many expressions can be cumbersome, knowing these identities expedites the process of evaluation, ensuring more accuracy while reducing complexity.