Problem 7 - Algebra for College Students

Question

Write each exponential statement in logarithmic form. For example, \(2^{5}=32\) becomes \(\log _{2} 32=\) 5 in logarithmic form. (c) \(\left(g^{-1} \circ f^{-1}\right)(x)\) \(\frac{x+7}{6}\) $$ 2^{-2}=\frac{1}{4} $$

Step-by-Step Solution

Answer

The exponential statement \(2^{-2} = \frac{1}{4}\) in logarithmic form is \(\log_{2}\left(\frac{1}{4}\right) = -2\).

Step 1: Understand the Exponential Statement

We start with the exponential statement provided in the exercise, which is \(2^{-2} = \frac{1}{4}\). This states that the base \(2\) raised to the power of \(-2\) equals \(\frac{1}{4}\).

Step 2: Identify Components for Logarithmic Form

In a logarithmic statement of the form \(\log_b y = x\), \(b\) is the base, \(y\) is the result, and \(x\) is the exponent. From \(2^{-2} = \frac{1}{4}\), \(b = 2\), \(y = \frac{1}{4}\), and \(x = -2\).

Step 3: Convert Exponential to Logarithmic Form

Substitute the identified components into the logarithmic form: \(\log_{2}(\frac{1}{4}) = -2\). This represents the logarithmic equivalent of the exponential equation \(2^{-2} = \frac{1}{4}\).

Key Concepts

Logarithmic Form

Logarithmic form is a way of expressing exponential relationships in terms of logarithms. If you have an exponential equation like \(a^b = c\), it can be converted into logarithmic form as \(\log_a c = b\). Here, \(a\) is the base, \(b\) is the exponent, and \(c\) is the result of the exponentiation.
Logarithms essentially answer the question: "To what exponent must the base be raised, to yield a specific number?"

Understanding this form is essential because it simplifies complex operations in mathematics, particularly when dealing with exponential growth or decay and in solving for unknowns in exponent calculations.

• Base \(a\): The number that is raised to a power in the exponential equation.
• Result \(c\): The outcome from the base raised to the exponent.
• Exponent \(b\): The power to which the base is raised in the exponential equation.

Getting familiar with logarithmic form allows easier manipulation of exponential equations and is widely used in various fields, such as biology, economics, and computer science.

Exponential to Logarithmic Conversion

Converting from an exponential form to logarithmic form can initially seem complex, but it follows a simple pattern. Consider the exponential equation \(2^{-2} = \frac{1}{4}\). To convert this to a logarithmic equation, identify the base, the exponent, and the result.

Here’s how this conversion works:

• The base \(b\) of logarithms is the same as the base in the exponential form.
• The result of the original exponentiation \(\frac{1}{4}\) becomes the number you are taking the log of.
• The exponent \(-2\) is what the logarithm equals.

Thus, \(2^{-2} = \frac{1}{4}\) becomes \(\log_2 \frac{1}{4} = -2\).
Conversion between these forms is crucial for simplifying and solving equations that have exponents. It’s like translating between two languages in math, enabling us to utilize logarithmic properties to solve complex problems effectively.

Powers and Exponents

Powers and exponents form the foundation of expressions that use repeated multiplication of a number. For example, \(a^b\) is the notation used to declare that \(a\) is multiplied by itself \(b\) times. When you see \(2^{-2}\), it means \(2\) is taken to the power of \(-2\), or \(\frac{1}{2^2}\), which equals \(\frac{1}{4}\).

Powers and exponents are essential in simplifying products of the same number into a more manageable form.

• Positive Exponents: Indicates straightforward multiplication, such as \(2^3 = 2 \times 2 \times 2 = 8\).
• Negative Exponents: Represent the reciprocal of the base raised to the corresponding positive exponent, such as \(2^{-2} = \frac{1}{2^2} = \frac{1}{4}\).
• Zero Exponent Rule: Any base raised to the power of zero equals \(1\), as in \(a^0 = 1\), assuming \(a eq 0\).

Powers and exponents are everywhere in the world of mathematics, from simple arithmetic to complex scientific calculations, making them a crucial tool for students to master.