Problem 22 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \log _{4} 4^{-1.23} $$

Step-by-Step Solution

Answer

-1.23

Step 1: Apply the Power Rule for Logarithms

The power rule for logarithms states that \( \log_b(a^c) = c \cdot \log_b(a) \). In this case, the expression is \( \log_4(4^{-1.23}) \), so we apply the power rule as follows:\[ \log_4(4^{-1.23}) = -1.23 \cdot \log_4(4) \]

Step 2: Simplify Logarithm of a Base to Itself

The property of logarithms \( \log_b(b) = 1 \) is used to simplify \( \log_4(4) \). Therefore:\[ \log_4(4) = 1 \]

Step 3: Substitute Back and Simplify

Substitute the result from Step 2 back into the expression from Step 1:\[ -1.23 \cdot \log_4(4) = -1.23 \cdot 1 \]Therefore, the simplified expression is:\[ -1.23 \]

Key Concepts

Power Rule

The power rule is a crucial concept in the field of logarithms that helps simplify expressions when dealing with exponents. It specifically states that if you have a logarithm of an exponential form, such as

• \( \log_b(a^c) \)

you can "bring down" the exponent as a multiplier to the entire logarithm. Therefore, the expression becomes:

• \( c \cdot \log_b(a) \)

To illustrate this, let's look at our given problem:

• \( \log_4(4^{-1.23}) \)

Using the power rule:

• \(-1.23 \cdot \log_4(4) \)

This transformation makes the expression easier to handle in subsequent steps as most characteristics of logarithms become evident. It highlights the importance of mastering the power rule for efficiently simplifying logarithmic expressions.

Logarithm Properties

Logarithm properties are a set of rules that can make manipulating logarithmic expressions much simpler. These properties are essential tools in solving many mathematical problems. One of the key properties used in this exercise is the idea that a logarithm of a number with the same base is equal to one:

• \( \log_b(b) = 1 \)

This property was applied directly in our example:

• \( \log_4(4) \)
• The base 4 raised to the power of 0 is 4, highlighting a base raised to 1 results in itself.

This significant property allows us to quickly simplify components of logarithmic expressions. By understanding and using these properties effectively, you can break down complex logarithms into clearer and simpler expressions.

Simplifying Logarithms

Simplifying logarithms often involves applying rules and properties to create a more straightforward expression. The example you provided showcases a typical process of simplification, including several important steps.
Initially, we leveraged the power rule to separate the exponent from the logarithmic function:

• \( -1.23 \cdot \log_4(4) \)

Afterward, a crucial step was simplifying the part of the expression that dealt with \( \log_4(4) \), which, as determined by properties of logarithms, equaled 1.
Putting it all together:

• \( -1.23 \times 1 = -1.23 \)

The result is a simple numeric value. Such simplification reduces expressions and makes them more interpretable, showing how intertwined logarithmic rules are with simplifying steps. Mastery of these techniques is essential for working comfortably with logarithms in any mathematical context.