Problem 104 - College Algebra and Calculus: An Applied Approach

Question

Find a logarithmic equation that relates \(y\) and \(x\). Explain the steps used to find the equation.$$ \begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 1 & 1.587 & 2.080 & 2.520 & 2.924 & 3.302 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The logarithmic equation that relates \(y\) and \(x\) in this case is \(y = \log_{1.587}x\).

Step 1: Find Base of Logarithm

To find the base of the logarithm, one may observe the relationship between \(x\) and \(y\), noticing the constant ratio between successive \(y\) values. In this case, that ratio is approximately 1.587/1 = 1.587. Thus, the base of logarithm, \(b\), is about 1.587.

Step 2: Choose one data pair and Write Logarithmic Equation

With the base discovered in step 1, you can use one of the data pairs to write a logarithmic equation. In this case, we could choose the first pair \(x = 1\) and \(y = 1\). So the equation is \(y = \log_b{x}\) or \(1 = \log_{1.587}{1}\).

Step 3: Test and Confirm the Equation

Testing if the base discovered in step 1 is correct is an important part of this problem. Test our equation with several pairs from the table. For example, when \(x = 3\), \(y \approx \log_{1.587}3\) which is approximately 2.080. If this works for multiple pairs, we can be confident our base is correct.

Key Concepts

Base of Logarithm

Understanding the base of a logarithm is crucial for solving logarithmic equations. The base of a logarithm, denoted as 'b' in the expression \(\log_b(x)\), shows the number which raised to the power of the logarithm results in 'x'. This relates to how many times one would have to multiply the base by itself to achieve the original number. For instance, if \(4 = \log_2(16)\), this means that \(2^4 = 16\).

In our example, the base was approximated by observing the ratio of successive 'y' values in a set of data. This ratio can often give a good estimation of the base if the data follows a logarithmic trend. In practice, identifying the correct base requires careful analysis of the pattern observed in the data. Once the base is identified, it can be used to construct and solve logarithmic equations that describe the relationship between the variables in the dataset presented.

Logarithmic Relationship

A logarithmic relationship occurs when one variable is a logarithm of another. This type of relationship is especially important in describing phenomena that grow exponentially or decay. In the context of our exercise, the process of finding the logarithmic relationship involves observing how the value of 'y' increases as 'x' changes.

For example, if 'y' grows quickly at the start but then its rate of increase slows down, it suggests an exponential growth where 'x' is being raised to a power to determine 'y'. The logarithm essentially reverses this process, allowing us to express 'x' as a function of 'y'. To put it simply, it answers the question: to what power must we raise the base to get the number 'x'?

Once we establish a consistent growth rate or pattern within the data, we can articulate a specific logarithmic relationship, typically formulated as \(y = \log_b(x)\) where 'b' is the base, 'x' is the value we are taking the logarithm of, and 'y' is the result or the power to which the base is raised.

Calculating Logarithms

• Identify the base of the logarithm from the pattern or given information.
• Select a pair of 'x' and 'y' values that exhibit the logarithmic relationship.
• Write the logarithmic equation using the observed base and the selected pair of values.
• Test the equation with other data pairs to confirm its accuracy.

When calculating logarithms, it is useful to know the laws of logarithms, such as the product, quotient, and power rules, which allow us to manipulate and simplify logarithmic expressions. In a more applied setting such as the one indicated in the solved problem, one can use a scientific calculator or software to find the value of logarithms to any base, which is essential for testing the logarithmic equation with various pairs of 'x' and 'y'.

With practice and awareness of these rules, calculating logarithms becomes an easier task, and it is a powerful tool for analyzing and interpreting exponential relationships in different areas of science and mathematics.