Problem 115 - College Algebra with Modeling and Visualization

Question

Change each equation to its equivalent exponential form. (a) \(\log _{8} x=3\) (b) \(\log _{9}(2+x)=5\) (c) \(\log _{k} b=c\)

Step-by-Step Solution

Answer

(a) \(8^3 = x\), (b) \(9^5 = 2 + x\), (c) \(k^c = b\).

Step 1: Identify the Components of the Logarithmic Equation (Part a)

The given equation is \(\log_{8} x = 3\). Here, the base is 8, the argument is \(x\), and the logarithm equals 3.

Step 2: Convert to Exponential Form (Part a)

The logarithmic equation \(\log_{8} x = 3\) can be written in exponential form as \(8^{3} = x\). This means that raising 8 to the power of 3 gives us \(x\).

Step 3: Identify the Components of the Logarithmic Equation (Part b)

The given equation is \(\log_{9}(2+x) = 5\). Here, the base is 9, the argument is \(2+x\), and the logarithm equals 5.

Step 4: Convert to Exponential Form (Part b)

The logarithmic equation \(\log_{9}(2+x) = 5\) can be converted to exponential form as \(9^{5} = 2+x\). This means raising 9 to the power of 5 results in \(2 + x\).

Step 5: Identify the Components of the Logarithmic Equation (Part c)

The given equation is \(\log_{k} b = c\). Here, the base is \(k\), the argument is \(b\), and the logarithm equals \(c\).

Step 6: Convert to Exponential Form (Part c)

The logarithmic equation \(\log_{k} b = c\) can be written in exponential form as \(k^{c} = b\). This means that raising \(k\) to the power of \(c\) gives \(b\).

Key Concepts

Logarithms

Logarithms are mathematical expressions that help us to solve equations involving exponential relationships. They are the inverse operations of exponentiation. For example, if you know the result of an exponential equation and want to find one of the variables, you can use a logarithm.
To understand a logarithmic expression such as \(\log_{b} a = c\):

• "\(b\)" is the base of the logarithm. It is the number that is raised to a power.
• "\(a\)" is the argument or result of the base raised to a power.
• "\(c\)" is the logarithm itself, which answers the question "to what power must the base be raised to produce \(a\)?"

When you see a logarithmic expression, it usually represents the power that a base must be raised to in order to get a certain number. This is very useful for reversing exponential equations, making them easier to solve. Transforming logarithmic expressions into exponential forms is a common practice, assisting in the comprehension and solution of mathematical problems.

Equation Transformation

Equation transformation is the process of converting one form of an equation into another. In mathematics, this allows us to manipulate equations to better solve or understand them.
For example, transforming a logarithmic equation into an exponential form requires a simple but careful application of mathematical principles. Using the given equation \(\log_{b} a = c\), we transform it to exponential form as \(b^{c} = a\).
Here’s the step-by-step process:

• Recognize the logarithmic equation, identifying the base, argument, and the value of the logarithm.
• Rearrange the equation into its exponential form.
• This helps us view the equation as \(b\) raised to the power of \(c\) equals \(a\).

This transformation allows us to use exponential equations for solving problems, such as finding unknown values in the argument or base. It is a key concept, especially in algebra and higher mathematics, where equations often change forms to reveal different aspects of a problem.

Mathematical Notation

Mathematical notation is a symbolic language used to convey mathematical concepts and processes succinctly and clearly. It enables us to express very complex ideas in a standardized form that can be understood globally.
A logarithm, for instance, is represented by \(\log\), usually followed by a subscript indicating the base, and a parenthetical or direct argument, showing the number involved. This standardization allows mathematicians and students worldwide to identify and solve logarithmic equations efficiently.
By writing \(\log_{b} a = c\), we convey a wealth of information:

• "\(b\)" indicates the base, showing what number is repeatedly multiplied.
• "\(a\)" is the result of raising the base to the "\(c\)" power.
• "\(c\)" represents the exponent or power itself.

Such notation not only saves space but also provides a clear, unambiguous depiction of mathematical relationships, essential for problem-solving and advancing mathematical theory. Understanding how to read and transform these symbols is crucial for success in mathematics.