Problem 10 - Precalculus with Calculus Previews

Question

In Problems \(7-12\), rewrite the given logarithmic expression as an equivalent exponential expression. $$ \log _{16} 2=\frac{1}{4} $$

Step-by-Step Solution

Answer

The equivalent exponential expression is \( 16^{\frac{1}{4}} = 2 \).

Step 1: Understand the logarithmic equation

The given logarithmic expression is \( \log_{16} 2 = \frac{1}{4} \). This means that the base 16 logarithm of 2 is equal to \( \frac{1}{4} \).

Step 2: Convert logarithmic to exponential form

The general form of a logarithmic equation \( \log_b a = c \) can be rewritten in exponential form as \( b^c = a \). Here, \( b = 16 \), \( c = \frac{1}{4} \), and \( a = 2 \).

Step 3: Write the exponential expression

Using the conversion formula from the previous step, we have \( 16^{\frac{1}{4}} = 2 \). This is the equivalent exponential expression for the given logarithmic equation.

Key Concepts

Understanding Exponential Expressions

Exponential expressions are mathematical expressions that involve numbers raised to a power or exponent. In such expressions, the number being multiplied is referred to as the "base," and the exponent indicates how many times the base is used as a factor. For example, in the expression \(3^4\), 3 is the base, and 4 is the exponent. This means 3 is multiplied by itself 4 times, resulting in \(3 \times 3 \times 3 \times 3 = 81\). Exponential expressions are fundamental in mathematics as they provide a concise way to represent repeated multiplication. They are powerful tools used in various fields such as science, engineering, and finance.

• Base: The number that is repeatedly multiplied is known as the base of the expression.
• Exponent: This indicates the number of times the base is multiplied by itself.
• Power: The expression as a whole, determined by both the base and the exponent (e.g., "power of 4").

Understanding exponential expressions involves recognizing these components and applying them to evaluate or transform mathematical equations.

Base of Logarithm

The concept of a base in logarithmic expressions is crucial. In any logarithmic expression, such as \( \log_b a = c \), the 'b' represents the "base of the logarithm." It signifies the number that is raised to a power (the exponent) to reach another number.For example, consider the expression \( \log_{16} 2 = \frac{1}{4} \). Here, 16 is the base. This means that 16 is the number that needs to be raised to the power \( \frac{1}{4} \) to result in 2. In simpler terms, you are asking "what power must 16 be raised to, to obtain 2?"Understanding this base concept is essential, as it acts as the cornerstone around which both exponential and logarithmic expressions revolve. It helps in:

• Transforming logarithmic expressions into exponential form.
• Determining the growth or decay factor, especially in real-world applications.
• Simplifying complex problems by focusing on the base and its exponent.

Thus, the base in a logarithmic function connects the question of power with the scale at which numbers are compared or evaluated.

Converting Logarithms to Exponents

Converting logarithmic expressions to exponential form is a vital skill in algebra that reveals the relationship between the base, the result of the logarithm, and the exponent. The key is understanding the fundamental formula: If you have a logarithmic expression like \( \log_b a = c \), you can convert it to an exponential form using: \[ b^c = a \]Let's take the example \( \log_{16} 2 = \frac{1}{4} \). According to our conversion rule:

• The base \(b\) is 16.
• The exponent \(c\) is \(\frac{1}{4}\).
• The result is 2, so we express it as: \(16^{\frac{1}{4}} = 2\).

This process involves understanding the inverse nature of logarithms and exponents. Logarithms answer the question of how many of one number (the base) multiply together to get another number. Conversely, exponentiation is the literal raising of a base to a power. By mastering this conversion, solving logarithmic problems becomes more intuitive and broadens your mathematical toolbox, making it easier to handle equations in both pure and applied mathematics.