Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Find the exact value of the logarithmic expression without using a calculator.\(\log _{4} \sqrt[3]{4}\)

Step-by-Step Solution

Answer

The exact value of the logarithmic expression \( \log _{4} \sqrt[3]{4} \) is \( \frac{1}{3} \).

Step 1: Rewrite the Radical as an Exponent

The cube root of a number can be expressed as that number raised to the power of \( \frac{1}{3} \). Therefore, \( \sqrt[3]{4} \) can be written as \( 4^{\frac{1}{3}} \). So, the expression we are dealing with is now \( \log_{4} 4^{\frac{1}{3}} \).

Step 2: Apply the power rule of logarithms

The power rule of logarithms states that \( \log_b a^n = n \cdot \log_b a \). Applying this to our expression, we get \( \frac{1}{3} \cdot \log_{4} 4 \).

Step 3: Evaluate \( \log_{4} 4 \)

By the definition of a logarithm, \( \log_{b} b = 1 \) for all \( b > 0 \), \( b \neq 1 \). Therefore, \( \log_{4} 4 = 1 \). Substitute this into the expression from Step 2, we're left with \( \frac{1}{3} \cdot 1 = \frac{1}{3} \).

Key Concepts

Logarithm Power Rule

Understanding logarithms can initially be challenging, but the logarithm power rule simplifies dealing with logarithms of exponential terms. This rule is a game-changer as it breaks down complex expressions into more manageable pieces.

The rule itself states that for any positive number a, base b (where b is also a positive number not equal to 1), and exponent n, the logarithm of a raised to the power of n, written as \( \log_b a^n \), can be rewritten as n times the logarithm of a with base b, expressed as \( n \cdot \log_b a \). Why is this useful? Well, it lets us pull out exponents so we can work with them directly, greatly simplifying the evaluation process.

Let's illustrate with an example:

• If you have \( \log_2(8^3) \), apply the power rule to get \( 3 \cdot \log_2 8 \), which then simplifies to \( 3 \cdot 3 = 9 \) since 2 cubed equals 8.

This principle directly assists students in tackling logarithmic problems by making it easier to see the connections between the logarithm's base and the argument's base.

Cube Root Exponentiation

When it comes to roots and powers, they're essentially two sides of the same coin. This is especially true for the cube root, which can seem less intuitive than the square root. Cube root exponentiation allows for expressing cube roots (\( \sqrt[3]{a} \) or the third root of a) in a more versatile way: as an exponent.

The concept is that any number's cube root can be shown as that number to the power of \( \frac{1}{3} \). So for any number a, \( \sqrt[3]{a} \)= \( a^{\frac{1}{3}} \). This is part of the general rule for roots, where the nth root of a is \( a^{\frac{1}{n}} \).

For example, if you need to express \( \sqrt[3]{8} \) in exponential form, you would write it as \( 8^{\frac{1}{3}} \), which is equal to 2 because two cubed is eight. This transformation is not just a neat trick; it opens the door to applying logarithmic rules, as seen in the exercise we're discussing.

Logarithmic Evaluation

When it comes to crunching numbers without a calculator, logarithmic evaluation can seem daunting. However, with some know-how, this process can be demystified. The foundational concept of logarithmic evaluation is understanding what a logarithm represents. The expression \( \log_b a \) answers the question: to what power must we raise b to get a?

If you know the base and its relationship to the argument, then evaluating the logarithm becomes a straightforward task. Remember:

• \( \log_b b = 1 \) because b to the power of 1 is b.
• \( \log_b 1 = 0 \) because any number to the power of 0 is 1.

In the context of our example \( \log_{4} 4 \) equals 1 because 4 to the power of 1 is 4. Thus, understanding simple integer exponents is critical in evaluating basic logarithms. Once you can evaluate simple logs, applying rules like the power rule becomes a powerful tool to simplify and solve even more complex logarithmic expressions.