Problem 35 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$\log _{4} 2$$

Step-by-Step Solution

Answer

\(\log_{4} 2 = \frac{1}{2}\).

Step 1: Understanding the logarithm

The expression \( \log_{4} 2 \) represents the logarithm of 2 with base 4. This asks the question: 4 raised to what power equals 2?

Step 2: Expressing the logarithm in terms of exponents

Set \( x = \log_{4} 2 \). This means \( 4^x = 2 \). We will use properties of exponents to solve for \( x \).

Step 3: Converting to a common base

Notice that both 4 and 2 can be expressed as powers of 2: \( 4 = 2^2 \). Substitute this into the equation so \( (2^2)^x = 2 \).

Step 4: Simplifying the equation

Using the law of exponents, \( (a^m)^n = a^{m } \), simplify the left side: \( 2^{2x} = 2^1 \).

Step 5: Equating the exponents

Since the bases are the same, equate the exponents: \( 2x = 1 \).

Step 6: Solving for x

Divide both sides by 2 to solve for \( x \): \( x = \frac{1}{2} \).

Key Concepts

Properties of Exponents

Exponents are powerful tools in mathematics that allow us to express repeated multiplication compactly. When dealing with logarithms and exponential expressions, it's essential to understand and utilize the properties of exponents effectively.

• **Product of Exponents**: This property states that when multiplying two exponents with the same base, you simply add the exponents. Mathematically, it’s expressed as: \( a^m \times a^n = a^{m+n} \).
• **Quotient of Exponents**: When dividing two exponents with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• **Power of a Power**: This property indicates that when an exponent is raised to another power, you multiply the exponents: \( (a^m)^n = a^{m\cdot n} \). This is the property used in simplifying \( (2^2)^x \) to \( 2^{2x} \).
• **Zero Exponent**: Any non-zero number raised to the zero power is 1: \( a^0 = 1 \).
• **Negative Exponent**: An exponent that is negative denotes a reciprocal: \( a^{-m} = \frac{1}{a^m} \).

Having a solid grasp of these properties can simplify complex expressions and solve equations efficiently. They lay the groundwork for handling exponential and logarithmic functions with ease.

Simplifying Expressions

Simplifying expressions often involves using properties of numbers and operations. In our exercise, the expression was \( \log_{4} 2 \). The aim was to simplify it step-by-step using logarithmic and exponential rules.
One of the key ways to simplify expressions is to express numbers in terms of a common base where possible. In this case, 4 and 2 were both expressed as powers of the number 2:

• 4, which is \( 2^2 \), helped us express \( 4^x \) in terms of base 2, making it easier to compare with 2 on the right-hand side.

Applying the power of a power property, \( (2^2)^x \) was simplified to \( 2^{2x} \).
This simplification step made it possible to directly compare the exponents. Because the bases were identical, it was straightforward to solve by setting the exponents equal: \( 2x = 1 \).
Breaking down expressions into equivalent forms often reveals solutions that seem more complicated at first glance. With practice, these methods become intuitive, making complex calculations less daunting.

Exponential Functions

Exponential functions are functions where the variable is in the exponent, typically written as \( f(x) = a^x \), where \( a \) is a constant.
These functions are characterized by rapid growth or decay and are foundational in various scientific fields, including physics, economics, and biology.
In the exercise, we essentially worked with an exponential function where we had to equate two sides: \( 4^x = 2 \).

• This was re-expressed using properties of exponents for easier comparison.
• The problem then boiled down to finding \( x \) such that \( a^x = b \), a typical question in logarithms and exponential functions.

The solution involved recognizing that both the left and right side could be converted to the same base, simplifying the process to merely equating exponents. This shows the practical application of understanding exponential functions and their solutions.
Exponential functions may seem complex, but with practice and understanding of how they behave and how they interact with logarithms, they become much more manageable. It's all about breaking down the parts and making the unfamiliar, familiar.