Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write each exponential equation in logarithmic form. $$ 2^{4}=16 $$

Step-by-Step Solution

Answer

\( \log_2(16) = 4 \)

Step 1: Understand the Problem

We are asked to rewrite an exponential equation in logarithmic form. The given equation embodies the form \( a^b = c \), where \( a \) is the base (2), \( b \) is the exponent (4), and \( c \) is the result (16).

Step 2: Identify the Logarithmic Form Structure

The logarithmic form of an exponential equation \( a^b = c \) is expressed as \( \log_a(c) = b \). This means we have to identify the components: \( a \) as the base of the logarithm, \( c \) as the input of the logarithm function, and \( b \) as the result of the logarithmic equation.

Step 3: Convert Exponential to Logarithmic Form

Using the relationship identified, we convert \( 2^4 = 16 \) into logarithmic form. We recognize \( a = 2 \), \( c = 16 \), and \( b = 4 \), which transforms to \( \log_2(16) = 4 \).

Key Concepts

Exponential Equations

Exponential equations play a crucial role in various fields including mathematics, science, and finance. They are equations where numbers undergo repeated multiplication, which can be represented compactly using exponents. In an exponential equation, you have a base raised to a power, or exponent, to yield a result. Let's consider the example given in the original exercise:

• The base is 2.
• The exponent is 4.
• The result is 16.

These equations are valuable for expressing rapidly growing or shrinking quantities. For example, exponential growth is often used to model populations or investments. Understanding how to manipulate and convert these equations is fundamental for more advanced math problems and applications.

Logarithms

Logarithms might seem intimidating at first, but they are incredibly useful tools for undoing exponentiation. Essentially, a logarithm answers the question: to what exponent must the base be raised to produce a given number? By transforming exponential statements into logarithmic form, we can solve for a variable that sits in the exponent position. This is especially useful in situations where the exponent is unknown, and the operation can't be easily reversed by simple arithmetic. A logarithm consists of:

• The base, the same as in the original exponential equation.
• The argument, which is the result of the exponential equation.
• The exponent or the logarithm value, which is what we're solving for.

In our example, \[ 2^4 = 16 \]converts to the logarithmic form, \[ \log_2(16) = 4 \]. This means that 2 raised to the power of 4 equals 16. Understanding this method of converting between forms is critical for tackling math problems that involve complexities like compound interest or sound intensity.

Converting Forms

Transitioning between exponential and logarithmic forms is a fundamental technique in solving equations. It allows for flexibility in approach and often simplifies complex calculations. The process involves understanding the core relationship between the base, exponent, and result in exponential equations, and translating it into a logarithmic statement.Here's a simple breakdown:

• Identify the base of the exponential equation.
• Determine the exponent, which will become the output of the logarithm.
• Recognize the result of the exponential equation, which will become the input (or argument) of the logarithm.

Taking the exponential form,\[ 2^4 = 16\],requires us to translate it into \[ \log_2(16) = 4 \]. This conversion underlines the relationship between these mathematical expressions. Practicing these conversions will strengthen your ability to move between different mathematical representations, enhancing your problem-solving skills across various topics.