Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in exponential form. \(\log _{y}(137)=x\)

Step-by-Step Solution

Answer

The exponential form is \(y^x = 137\).

Step 1: Understand the Logarithmic Form

The logarithmic equation provided is \(\log_y(137) = x\), which signifies that \(y\) is the base of the logarithm, \(137\) is the number, and \(x\) is the exponent.

Step 2: Rewrite in Exponential Form

To convert the logarithmic equation \(\log_y(137) = x\) into its exponential form, we use the basic definition of a logarithm: if \(\log_b(N) = M\), then \(b^M = N\). Applying this, we get \(y^x = 137\).

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that relate three components: a base, an exponent, and a number. They help simplify and solve problems where exponential growths or scales are involved. In the given equation \(\log_y(137) = x\), the base is \(y\), which is raised to the power \(x\) to result in 137. Understanding the components and how they relate is fundamental to interpreting these equations.

To understand logarithmic equations better, grasp the following key points:

• The base of the logarithm ('\(y\)' in this case) is the number that the exponent will be applied to.
• The result (here, 137) is what you get after raising the base to the given power or exponent ('\(x\)' in this equation).
• Logarithmic equations are the inverse operations of exponential equations.

Recognizing how these variables interact will allow you to manipulate and solve logarithms effectively.

Exponential Equations

Exponential equations involve an expression where a constant base is raised to a variable exponent. They are written in the form \(b^x = N\), and they model situations where numbers grow or decay exponentially. When converting from the logarithmic form \(\log_y(137) = x\) to the exponential form \(y^x = 137\), the operation becomes more straightforward.

Main aspects of exponential equations include:

• The base ('\(y\)' in our example) is a fixed number that gets raised to the power of the exponent.
• The exponent ('\(x\)' in this context) determines how many times the base multiplies by itself.
• The result (137) is known as the power, representing the true magnitude resulting from the operation.

Exponential equations are powerful tools in scenarios involving growth or decay, like population studies, radioactive decay, and interest calculations. Mastering them is crucial for solving real-world mathematical problems.

Basic Definition of Logarithm

The basic definition of a logarithm provides a direct way to convert between logarithmic and exponential forms. A logarithm answers the question: "To what exponent must the base be raised to achieve a given number?" In this context, the question becomes "To what power must \(y\) be raised to result in 137?"

Understanding the fundamental definition:

• If \(\log_b(N) = M\), then it translates to \(b^M = N\).
• Essentially, the logarithm finds the exponent in the related exponential equation.
• The base \(b\) is assumed to be positive and not equal to one to keep the operation meaningful and defined.

This definition is the foundation for more advanced mathematical concepts, allowing you to derive one form from another effectively. Getting familiar with this concept will make it easier to switch between different mathematical expressions.