Problem 28 - College Algebra

Question

Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{4}$$

Step-by-Step Solution

Answer

The short answer to this exercise is \(-2 / \log_{2} 3\). This is the most simplified form of the expression, as \(\log_{2} 3\) can't be simplified without a calculator.

Step 1: Expression of the Fraction in Power Form

Firstly, the fraction \(\frac{1}{4}\) can be expressed as \(4^{-1}\), as any fraction \(1/n\) is equivalent to \(n^{-1}\). Now, \(4\) can be expressed as \(2^{2}\), so the expression becomes \( (2^2)^{-1}\). Therefore, the whole expression becomes \(2^{-2}\).

Step 2: Change the Logarithm Base

As the fraction is now in terms of \(2\), it would be helpful to change the base of the logarithm to \(2\). This can be achieved by using the formula for change of base, where \(\log_b a = \frac{\log_d a}{\log_d b}\). Applying this to our expression gives us \(\log_{2} 2^{-2} / \log_{2} 3\).

Step 3: Evaluate the Expression

Now, we evaluate \(\log_{2} 2^{-2}\), which is simply \(-2\). \(\log_{2} 3\) can't be simplified further. So, our final expression is \(-2 / \log_{2} 3\).

Key Concepts

Change of Base

The change of base formula is a handy tool when dealing with logarithms, especially when you want to convert a logarithm to a different base. This can be particularly useful when the base is not easily workable, as in our original exercise with base 3. Changing the base allows you to solve the expression with what's on hand.

The formula for change of base is given by:

• \( \log_b a = \frac{\log_d a}{\log_d b} \)

Here, you're essentially converting the logarithm from base \( b \) to a new base \( d \). This conversion is useful when your new base \( d \) is closer or simpler to work with, such as base 10 or base 2.

For example, using the change of base formula in our exercise, we transform \( \log_3 \frac{1}{4} \) to \( \frac{\log_2 2^{-2}}{\log_2 3} \). This makes further evaluations easier.

Fractional Logarithms

Logarithms with fractional values can be intimidating, but understanding them can significantly ease calculations. In our expression, \( \log_3 \frac{1}{4} \), we see a fraction \( \frac{1}{4} \). To manage this, you must express the fraction in terms of powers.

Any fraction \( \frac{1}{n} \) can be expressed as \( n^{-1} \). This transforms the problem from dealing with division to dealing with exponents. Let's use the specific case of \( \frac{1}{4} \):

• First, convert \( 4 \) into its power form: \( 4 = 2^2 \).
• Thus, \( \frac{1}{4} = (2^2)^{-1} = 2^{-2} \).

By representing the fraction as a power, we have merely shifted the difficulty from dealing with a fraction to dealing with an exponent, which becomes manageable with logarithms.

Power of a Number

Understanding the concept of powers of a number is vital in simplifying logarithmic expressions, particularly when fractions are involved. The exponents provide a framework to rewrite numbers in a more workable form.

In our exercise, changing \( \frac{1}{4} \) into \( 2^{-2} \) through powers is a crucial step. This conversion is facilitated by understanding exponent rules:

• A base raised to a negative exponent means you are taking the reciprocal of the base raised to the positive exponent, e.g., \( a^{-b} = \frac{1}{a^b} \).

Additionally, when evaluating a logarithm where the base matches the number within the logarithm, such as \( \log_2 2^{-2} \), you simplify directly to the exponent, providing the solution \( -2 \) in this case.

This simplification drastically reduces the complexity of our original expression, making the problem less about tricky arithmetic and more about understanding how numbers interrelate through powers and logarithms.