Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in exponential form. \(\log _{4}(q)=m\)

Step-by-Step Solution

Answer

The exponential form is \( 4^m = q \).

Step 1: Understanding the Logarithmic Equation

The given equation is in logarithmic form: \( \log _{4}(q)=m \). In this form, it represents a relationship where 4 is the base, \( q \) is the result, and \( m \) is the exponent. Our task is to rewrite this logarithmic equation in its equivalent exponential form.

Step 2: Rewriting in Exponential Form

In exponential form, the equation \( \log _{b}(a) = c \) translates to \( b^c = a \). Applying this rule to our specific equation \( \log _{4}(q)=m \), the base is 4, the exponent is \( m \), and the result is \( q \). Therefore, the equation becomes \( 4^m = q \).

Key Concepts

Exponential Form

In mathematics, exponential form is a way to express repeated multiplication of the same number. It involves using a specific base number raised to a power, or exponent, to signify how many times the base is multiplied by itself. For instance, when faced with the expression \( a^n \), the \( a \) is the base, and the \( n \) indicates the number of times the base is used as a factor. In essence, this means multiplying the base \( a \) by itself \( n \) times.
Exponential form is particularly useful because it offers a simplified way of expressing large numbers and complex calculations. Instead of writing out an extensive multiplication, you can use exponentiation to denote the same operation. For example, \( 10^3 \) is a concise way to express \( 10 \times 10 \times 10 \), which equals 1000.
Understanding exponential form is essential when converting logarithmic equations. Logarithmic equations often describe the inverse relationships of exponential functions, which makes translating between these forms an important skill in algebra and calculus.

Logarithmic Equation

A logarithmic equation is an equation that involves a logarithm with an unknown variable. These equations represent a different way to express exponential relationships. The fundamental concept of a logarithm is that it answers the question: "To what power must a specific base be raised, to yield a particular number?"
The general form of a logarithmic equation is \( \log_b(a) = c \), where \( b \) is the base, \( a \) is the number, and \( c \) is the exponent or the power. This equation states that \( b \) raised to the power \( c \) equals \( a \). For example, if \( \log_{10}(100) = 2 \), then \( 10^2 = 100 \).
These equations are used widely across mathematics and applied sciences, especially when dealing with exponential growth or decay, such as population growth, radioactive decay, or even in finance through compound interest calculations. Understanding how to interpret and rewrite logarithmic equations in exponential form enables better interpretation and simplification of complex problems.

Base and Exponent

The terms 'base' and 'exponent' are key components in understanding both exponential and logarithmic expressions.

• The base is the number that is repeatedly multiplied in an exponential expression. In a logarithmic equation \( \log_b(a) = c \), the \( b \) represents the base of the logarithm. This number is crucial because it determines the scale and rate of growth represented by the equation.
• The exponent, on the other hand, indicates how many times the base is multiplied by itself. In exponential notation, like \( a^n \), the \( n \) is the exponent. In the context of a logarithmic equation, the exponent provides the power to which the base must be raised to give the original number. For \( \log_b(a) = c \), \( c \) is the exponent that leads to the value \( a \) when the base \( b \) is used.

These two components are interlinked with each other in expressions and equations. By understanding their roles, one can easily navigate through problems involving growth rates, indices, and many scientific applications. Whether dealing with exponential growth in biology or the pH scale in chemistry, the concepts of base and exponent make complicated calculations manageable.