Problem 43 - College Algebra

Question

For the following exercises, evaluate the base \(b\) logarithmic expression without using a calculator. $$\log _{6}(\sqrt{6})$$

Step-by-Step Solution

Answer

\( \log_{6}(\sqrt{6}) = \frac{1}{2} \).

Step 1: Identify the Expression

We are given the expression \( \log_{6}(\sqrt{6}) \) and need to simplify it. The term \( \sqrt{6} \) is the same as \( 6^{1/2} \).

Step 2: Apply Logarithmic Identity

Recall the identity \( \log_{b}(a^{c}) = c \cdot \log_{b}(a) \). Using this, we can rewrite the given expression: \( \log_{6}(6^{1/2}) = \frac{1}{2} \cdot \log_{6}(6) \).

Step 3: Evaluate the Logarithm

Since \( 6 \) to the power of 1 is 6, \( \log_{6}(6) = 1 \) by the identity \( \log_{b}(b) = 1 \).

Step 4: Simplify the Expression

Substitute \( \log_{6}(6) = 1 \) into the expression from Step 2: \( \frac{1}{2} \cdot 1 = \frac{1}{2} \). Thus, \( \log_{6}(\sqrt{6}) = \frac{1}{2} \).

Key Concepts

Logarithmic Expressions

Logarithmic expressions are a cornerstone in various fields of math and science. They help us express and solve equations involving powers. A logarithmic expression typically looks like this: \( \log_{b}(a) \), where \( b \) is the base, and \( a \) is the number you are taking the logarithm of. The value of a logarithmic expression answers the question, "To what power must \( b \) be raised to result in \( a \)?" For example, \( \log_{10}(100) \) equals 2, because 10 raised to the power of 2 is 100.

It's essential to understand that different bases change the context of the logarithm. The base-10 logarithms, known as common logarithms, are widely used. Meanwhile, base-\( e \) logarithms, often written as \( \ln \), are natural logarithms and frequently pop up in calculus. In the exercise \( \log_{6}(\sqrt{6}) \), the base is 6, indicating we must consider powers of 6 to solve it.

Logarithmic Identities

Logarithmic identities simplify our work with logarithms by allowing us to transform expressions. One critical identity used frequently is \( \log_{b}(a^{c}) = c \cdot \log_{b}(a) \). This property lets us bring exponents in the logarithm down in front as a multiplier.

Consider the problem \( \log_{6}(\sqrt{6}) \). The square root is equivalent to raising a number to the power of \( \frac{1}{2} \). Thus, we rewrite the expression as \( \log_{6}(6^{1/2}) \). Using our identity, it becomes \( \frac{1}{2} \cdot \log_{6}(6) \).

• Product Property: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \).
• Quotient Property: \( \log_{b}(\frac{m}{n}) = \log_{b}(m) - \log_{b}(n) \).
• Power Property: \( \log_{b}(m^{n}) = n \cdot \log_{b}(m) \).

These identities offer powerful tools to break down or aggregate complex logarithmic expressions seamlessly.

Simplifying Expressions

Simplifying logarithmic expressions involves reducing them to their simplest form. This often means applying identities, evaluating known values, and reducing terms. In our exercise, we simplified \( \log_{6}(\sqrt{6}) \) by reinterpreting \( \sqrt{6} \) as \( 6^{1/2} \), which allowed us to pull out the fractional exponent using the power property.

Next, knowing that \( \log_{6}(6) = 1 \) because any number raised to the power of 1 equals itself, helped reduce the expression further. Thus \( \frac{1}{2} \cdot \log_{6}(6) \) simplifies to \( \frac{1}{2} \cdot 1 \), which equals \( \frac{1}{2} \).
Here are some key tips for simplifying:

• Identify and apply relevant logarithmic identities.
• Substitute known logarithmic values like \( \log_{b}(b) = 1 \).
• Look for opportunities to rewrite expressions, such as turning roots into fractional exponents.

Simplifying expressions not only makes solving problems faster but also helps in understanding the underlying math concepts better.