Problem 19 - Intermediate Algebra

Question

a. Use a calculator to complete the table of values for \(f(x)=\log x .\) Round to the nearest hundredth. b. Graph \(f(x)=\log x .\) Note that the units on the \(x\) - and \(y\) -axes are different. $$ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 0.5 & \\ 1 & \\ 2 & \\ 4 & \\ 6 & \\ 8 & \\ 10 & \\ \hline \end{array} $$ GRAPH CANNOT COPY

Step-by-Step Solution

Answer

Log values: \(-0.30, 0, 0.30, 0.60, 0.78, 0.90, 1\). Graph is a rising curve.

Step 1: Calculate the Value for x = 0.5

Using a calculator, compute the logarithm base 10 for 0.5: \( f(0.5) = \log_{10} 0.5 \). This yields \( f(0.5) \approx -0.30 \) when rounded to the nearest hundredth.

Step 2: Calculate the Value for x = 1

Evaluate \( \log_{10} 1 \) using a calculator. Since the logarithm of 1 is always 0, we have \( f(1) = 0 \).

Step 3: Calculate the Value for x = 2

Determine the logarithm base 10 for 2: \( f(2) = \log_{10} 2 \). This results in \( f(2) \approx 0.30 \) when rounded to the nearest hundredth.

Step 4: Calculate the Value for x = 4

Compute \( \log_{10} 4 \) using a calculator, giving \( f(4) \approx 0.60 \) after rounding to the nearest hundredth.

Step 5: Calculate the Value for x = 6

Using a calculator, find \( \log_{10} 6 \) which is approximately \( f(6) \approx 0.78 \) when rounded to two decimal places.

Step 6: Calculate the Value for x = 8

Calculate \( \log_{10} 8 \), resulting in \( f(8) \approx 0.90 \) rounded to the nearest hundredth.

Step 7: Calculate the Value for x = 10

Since \( \log_{10} 10 = 1 \), we have \( f(10) = 1 \).

Step 8: Graph the Function

The values calculated are: \( f(0.5) = -0.30, \ f(1) = 0, \ f(2) = 0.30, \ f(4) = 0.60, \ f(6) = 0.78, \ f(8) = 0.90, \ f(10) = 1 \). Plot these points on a graph, noting different scales on the x-axis and y-axis due to differing unit sizes.

Step 9: Complete the Table and Sketch Graph

Fill the table with calculated values and sketch the curve rising to the right starting at a very negative value and asymptotic as it approaches the y-axis.

Key Concepts

Table of Values

Creating a table of values for a function helps understand its behavior by displaying input-output pairs. In our case, we observe this for the function \( f(x) = \log x \), specifically in base 10.Let's consider the function values at specified points:

• \(x = 0.5\): Calculating \(\log_{10} 0.5\), we find \(f(0.5) \approx -0.30\). This means the value is negative as 0.5 is less than 1.
• \(x = 1\): \(\log_{10} 1 = 0\). Since logarithm is an exponent, this result confirms that 10 raised to the 0 power is 1.
• \(x = 2\): For \(\log_{10} 2\), we get \(f(2) \approx 0.30\). This signifies only a slight increase, since 2 is close to 1.
• \(x = 4\): Calculating \(\log_{10} 4\), the result is \(f(4) \approx 0.60\), illustrating gradual growth.
• \(x = 6\): For \(\log_{10} 6\), we compute \(f(6) \approx 0.78\), further demonstrating this increase.
• \(x=8\): \(\log_{10} 8\) gives \(f(8) \approx 0.90\), showing that as \(x\) increases, \(f(x)\) also goes up.
• \(x = 10\): As already known, \(\log_{10} 10 = 1\), confirming the property of logarithms where the base itself results in 1.

A useful tip: when filling in these values, always round to the nearest hundredth for precision and uniformity.

Graphing Functions

Graphing functions like \( f(x) = \log x \) paints a visual picture of the function's behavior. This aids in understanding the nature and the flow of mathematical relationships.When plotting the function using the table of values:- Start by surveying the points: \( (0.5, -0.30), (1, 0), (2, 0.30), (4, 0.60), (6, 0.78), (8, 0.90), (10, 1) \).- Remember, as \( x \) gets larger, \( f(x) \) increases but at a decreasing rate.- Notice the unique shape of a logarithm graph: the curve starts from the negative x-values and climbs upwards to the right.- The x-axis will have larger units compared to the y-axis which frequently has smaller increments, due to its slow increase.- Notice the asymptotic behavior as the curve approaches the y-axis but never touches it, reflecting the domain condition \( x > 0 \).This graphing approach reflects the true nature of logarithms, displaying how growth slows with larger x-values.

Logarithms

Logarithms are a concise way to express rates of change or growth in mathematical expressions. They are inverse operations to exponents.Key insights:

• The base 10 logarithm, \( \log_{10} \), signifies what power a base number must be raised to yield a particular number.
• The concept of \( \log_{10} x = y \) means \( 10^y = x \).
• For values \( x < 1 \), the logarithms are negative. This reflects numbers that produce fractions when base 10 is raised to a power.
• At \( x = 1 \), logarithms return zero, indicating any number raised to the zero power results in 1.
• As values of \( x \) increase above 1, logarithms yield positive results, acknowledging the exponential growth of numbers.

Understanding logarithms requires envisioning how exponential operations dominate numerical behavior, thus providing a versatile tool in fields ranging from science to engineering.