Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in logarithmic form. \(y^{x}=\frac{39}{100}\)

Step-by-Step Solution

Answer

\( \log_{y}\left(\frac{39}{100}\right) = x \)

Step 1: Understanding the Initial Equation

We start with the equation given in the exponential form: \[ y^{x} = \frac{39}{100} \] Our task is to rewrite this equation in its equivalent logarithmic form.

Step 2: Applying the Definition of Logarithms

To convert an exponential equation of the form \(a^{b}=c\) into its logarithmic form, we use the definition of logarithms. It becomes \(\log_{a}(c) = b\). Here, \(y\) is the base, \(x\) is the exponent, and \(\frac{39}{100}\) is the result. Therefore, the equation becomes: \[ \log_{y}\left(\frac{39}{100}\right) = x \]

Key Concepts

Exponential Equations

Exponential equations tell us how many times we multiply the base by itself to reach another number. Let's consider the general form: \[ a^{b} = c \]Here, \(a\) is called the base, \(b\) is known as the exponent, and \(c\) is the result. The equation we've worked with, \(y^{x} = \frac{39}{100}\), is an example of an exponential equation. In it, \(y\) needs to be multiplied by itself \(x\) times to equal \(\frac{39}{100}\).

• Base: The number being multiplied.
• Exponent: The number of times the base is used as a factor.
• Result: What you get after multiplying the base by itself the exponent number of times.

Understanding exponential equations helps us in growth models, like population growth or compound interest, where values change quickly over time. In mathematical problems, we often need to transform these into other forms, such as logarithmic forms, to solve them easily.

Logarithmic Form Transformation

Turning an exponential equation into a logarithmic form is a vital technique. It allows us to solve equations where the unknown is an exponent. If we take an exponential equation, such as:\[ y^{x} = \frac{39}{100} \]We can change it into a logarithmic form by understanding that the logarithm finds the exponent that our base needs to reach our result. According to the definition of logarithms, it becomes:\[ \log_{y}\left(\frac{39}{100}\right) = x \]This transformation follows from the rule:\[ a^{b} = c \quad \Rightarrow \quad \log_{a}(c) = b \]With this conversion:

• We know "\(y\)" is the base.
• The input "\(\frac{39}{100}\)" is the result from the exponential form.
• "\(x\)" becomes the value our logarithm calculates.

Using logarithmic form makes handling complex problems, including solving for exponents, more manageable and understandable.

Base and Exponent Concepts

The base and exponent in exponential functions play a crucial role in how values behave. The base, typically positive, indicates what main number is repeatedly multiplied.To comprehend these components better:

• The Base, represented as \(a\), shows the recurring number in repeated multiplication.
• The Exponent, or \(b\), demonstrates how frequently the base is multiplied by itself.

For example, if we have \(2^{3}\), it means multiplying 2 by itself 3 times: \(2 \times 2 \times 2 = 8\). In an equation like \(y^{x} = \frac{39}{100}\), you determine the needed exponent "\(x\)" for the base "\(y\)" to reach \(\frac{39}{100}\).Understanding bases and exponents helps in breaking down the complexity of exponential calculations, providing insights into how numbers can grow or shrink exponentially, which is important for fields like finance or science.

Algebra Problem-Solving

Algebra is a dynamic toolset for solving equations, often involving finding unknown values. In cases dealing with exponential equations, our goal is often to solve for the exponent, which requires strategic transformations.Consider solving when the equation starts as:\[ y^{x} = \frac{39}{100} \]Here’s a quick guide to problem-solving with algebra:

• Identify the Unknown: Determine which part of your equation needs finding. In this context, it’s the exponent \(x\).
• Use Proper Transformation: Rewriting an equation to a logarithmic form can help isolate the variable of interest. For example, transforming to \(\log_{y}\left(\frac{39}{100}\right) = x\) allows us to express \(x\) directly.
• Solve Systematically: Use known rules and techniques like properties of logarithms to find concrete solutions.

Using these steps can demystify solving algebraic and exponential problems, making even complex equations simpler. It helps when approached logically and methodically.