Question

Find a logarithmic equation that relates \(y\) and \(x\). Explain the steps used to find the equation.$$ \begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 2.5 & 2.102 & 1.9 & 1.768 & 1.672 & 1.597 \\ \hline \end{array} $$

Short answer

The formula representing the logarithmic relationship between \(x\) and \(y\) is \(y = -0.253 \log_{10}(x) + 2.586\).

Step-by-step solution

Observe the Pattern

Looking at the values provided in the table, notice that as \(x\) increases, \(y\) decreases. This suggests a logarithmic relationship. Use this knowledge to make an initial educated guess of the form \(y=a \log_b(x)+c\), where \(a\), \(b\), and \(c\) are constants.

Calculate the constants

Assume the logarithm base to be 10 (meaning \(b=10\)). Try to solve for \(a\) and \(c\) using two points from the table. We can choose \((x, y) = (4, 1.768)\) and \((x, y) = (5, 1.672)\). This gives two equations, \(1.768 = a \log_{10}(4) + c\) and \(1.672 = a \log_{10}(5) + c\). By solving this system of equations, the values of \(a\) and \(c\) can be obtained, which is approximately -0.253 for \(a\) and 2.586 for \(c\).

Verify with other points

Using the obtained values for \(a\) and \(c\), check if the equation holds for other points given in the table. This will confirm if the equation accurately represents the relationship between \(x\) and \(y\).

Finalize the equation

Once verified, finalize the equation. If verification fails at any point, reconsider the choice of logarithm base or reassess the original assumption that the relationship is logarithmic.

Key concepts

Logarithm Base

Understanding the concept of logarithm base is vital when dealing with logarithmic equations. A logarithm tells us the power to which a number must be raised to obtain another number. For example, the logarithm of 1000 to the base 10 is 3, written as \(\log_{10}1000 = 3\), because \(10^3 = 1000\).

When solving logarithmic equations, the base of the logarithm plays a crucial role in how the equation is interpreted and solved. The base of a common logarithm, like the one assumed in the exercise, is 10; however, natural logarithms have a base of \(e\), where \(e\) is an irrational number approximately equal to 2.71828. If a specific base is not provided, it is often assumed to be 10 or \(e\), depending on the context. This standardization simplifies calculations in many real-world applications, such as exponential growth, sound intensity (decibels), and pH chemistry calculations.

In the given exercise, assuming the logarithm base to be 10 leads to determining the constants for the equation, based upon the data points provided. The choice of logarithmic base impacts the values of the constants in the equation and hence, its final form.

Solving System of Equations

Solving a system of equations is a foundational technique in mathematics that allows for determining the values of multiple variables. In practical terms, it involves finding the intersection points of various algebraic equations. When you have two or more equations working in tandem, you're looking for the specific values of variables that satisfy all the equations in the system simultaneously.

To solve a system of linear equations, such as the one seen in the logarithmic problem provided, you can use methods like substitution, elimination, or matrix operations. In the provided step-by-step solution, two equations are set up based on two sets of \(x, y\) coordinates. The method applied could be the substitution method wherein one variable is expressed in terms of the other, or the elimination method, where one variable is eliminated by adding or subtracting the equations.

Example of Using the Elimination Method:

By manipulating the two equations from the example, \(1.768 = a \(\log_{10}(4)\) + c\) and \(1.672 = a \(\log_{10}(5)\) + c\), one could subtract one equation from the other to eliminate \(c\) and solve for \(a\). Following this, they would back-substitute to find \(c\). The result offers insights into the system's characteristics and, in this exercise, the logarithmic relationship between \(x\) and \(y\).

Analyzing Data Patterns

Analyzing data patterns is a critical step in various scientific and mathematical fields, as it allows for the understanding and prediction of behavior based on observed data. In the context of logarithmic equations, recognizing a pattern in a set of data points can suggest the type of relationship that exists between variables.

When given a data table, as in the exercise, one should first look for a trend or relationship by plotting the data if necessary. A decreasing pattern as seen with increasing \(x\) and decreasing \(y\) values can indicate a logarithmic relationship, suggesting that as one variable increases, the other decreases according to a logarithmic curve.

Once a potential logarithmic pattern is identified, as it was in the step-by-step solution, we then apply the formula for a logarithmic function and use known data points to calculate unknown constants. This leads to the formulation of a model that describes the observed data. It is important to verify the model against other data points to ensure its accuracy and reliability. If the model doesn't fit well, reconsideration of the chosen function type, or the inclusion of additional variables or transformations may be necessary.