Problem 23 - College Algebra

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: Understand the Relationship Between Exponential and Logarithmic Forms

The exponential form of an equation is typically written as \( b^x = y \). The corresponding logarithmic form is \( \log_b(y) = x \). In this exercise, we have to rewrite the given exponential equation in its logarithmic form.

Step 2: Identify the Base, Exponent, and Result in the Exponential Equation

In the given equation \( y^x = \frac{39}{100} \), you need to identify: the base \( b = y \), the exponent \( x \), and the result \( \frac{39}{100} \).

Step 3: Rewrite the Equation in Logarithmic Form

Following the structure of \( \log_b(y) = x \), we substitute the identified parts: base \( b = y \), result \( \frac{39}{100} \), and exponent \( x \). Therefore, the logarithmic form is \( \log_y\left(\frac{39}{100}\right) = x \).

Key Concepts

Exponential Form

The exponential form of an equation is a way of expressing numbers using a base and an exponent. It is commonly written as \( b^x = y \), where:

• \( b \) is the base
• \( x \) is the exponent
• \( y \) is the result or the power

This form is used to represent repeated multiplication of the base \( b \). For example, in the equation \( 2^3 = 8 \), the base \( 2 \) is multiplied by itself three times to get 8.
Exponential forms are quite useful in various areas of mathematics and science, helping us simplify calculations and understand exponential growth or decay.

Base of Logarithm

The base of a logarithm is a critical part of understanding logarithmic expressions. It is the number that gets raised to a certain power in the corresponding exponential form. For example, in the logarithmic expression \( \log_b(y) = x \), \( b \) is the base.

• It must always be a positive number and cannot be 1.
• The base signifies the number of times it should be multiplied by itself to reach the result \( y \) in the exponential form.

In our original exercise, the base \( b \) is \( y \) in the equation \( y^x = \frac{39}{100} \), highlighting how the base operates within both forms.

Exponents

Exponents, also known as powers, are integral in understanding exponential equations. An exponent tells you how many times to multiply the base by itself. In any expression of the form \( b^x \), \( x \) is the exponent.
It can be a positive integer, zero, or a negative integer, each modifying the base in specific ways:

• Positive exponents imply repeated multiplication.
• Zero exponents mean the result is always 1, regardless of the base.
• Negative exponents denote division or reciprocation of the base.

In the exercise equation \( y^x = \frac{39}{100} \), \( x \) is the exponent, demonstrating the repeated application of the base itself in the equation's context.

Conversion Between Exponential and Logarithmic Forms

Converting between exponential and logarithmic forms is a valuable skill in mathematics. It allows you to switch perspectives between different but related mathematical concepts.
The conversion process is straightforward:

• If you have an expression in exponential form, \( b^x = y \), convert it to logarithmic form by rearranging it to \( \log_b(y) = x \).
• Conversely, from a logarithmic form \( \log_b(y) = x \), you go back to an exponential form, \( b^x = y \).

This transformation is not just a matter of symbols. It provides deeper insight into the relationships of numbers in equations. In the original step-by-step solution, this conversion was used to find that \( \log_y\left(\frac{39}{100}\right) = x \), illustrating how the exponent and result from the exponential form can directly translate into the logarithmic form.