Problem 1 - Precalculus

Question

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(2^{3}=8\)

Step-by-Step Solution

Answer

The equation \(2^3 = 8\) is \(\log_2(8) = 3\) in logarithmic form.

Step 1: Identify the Base, Exponent, and Result

In the given exponential equation, identify the base, which is 2, the exponent, which is 3, and the result, which is 8.

Step 2: Apply the Logarithmic Property

Using the property from Theorem 6.2, convert the exponential equation to a logarithmic form. This property states that if you have an equation in the form of \( b^a = c \), it can be rewritten as \( \log_b(c) = a \). For the equation \( 2^3 = 8 \), rewrite it as \( \log_2(8) = 3 \).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a constant base is raised to a variable exponent. The general form is given by \( b^x = y \), where \( b \) is the base, \( x \) is the exponent, and \( y \) is the result. These types of equations are quite common in real-world scenarios such as calculating compound interest or population growth.

To solve an exponential equation, you typically need to rewrite it in a form that allows you to determine the value of the unknown exponent. This frequently involves utilizing logarithms to transform the equation into a more manageable form. The exponential equation \( 2^3 = 8 \) is a simple example where the base is 2, the exponent is 3, and the result is 8. By understanding exponential equations, you can address more complex problems involving variables and constants.

Logarithmic Equations

Logarithmic equations involve finding the exponent to which a base number is raised to produce a given number. The general form of a logarithmic equation is \( \log_b(c) = a \), which asks: "to what power must \( b \) be raised, to give \( c \)?"

Logarithmic equations are closely tied to exponential equations. They are essentially the inverse operation of exponentiation. When dealing with logarithmic equations, it is crucial to understand that they help in solving equations where the unknown is the exponent. For instance, if you have the logarithmic equation \( \log_2(8) = 3 \), it means that 2 raised to the power 3 results in 8. This connection makes logarithmic equations vital in various fields like earthquake measurement (Richter scale) and sound intensity (decibels).

Converting from an exponential form to a logarithmic form, and vice versa, is a powerful technique to solve for variables in both simple and complex mathematical contexts.

Theorem 6.2

Theorem 6.2 provides a pivotal bridge between exponential and logarithmic equations. It asserts that an equation in the form \( b^a = c \) can be equivalently expressed as \( \log_b(c) = a \). This theorem lays the foundation for converting between these two forms.

Utilizing Theorem 6.2 allows you to seamlessly shift between exponential and logarithmic formats, which is particularly useful for solving problems that involve growth and decay. It simplifies the task of isolating variables, especially when exponents are involved, and offers an intuitive method to interpret exponential relationships by viewing them through the lens of logarithms.

In practice, applying Theorem 6.2 efficiently transforms complex exponential relationships into straightforward logarithmic equations. This theorem is fundamental in fields such as finance, physics, and engineering where such transformations are not just useful but essential.