Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. (Value) $$x=345$$ $$x=\frac{4}{5}$$ $$x=14.8$$ $$x=4.3$$ (Function) $$h(x)=1.9 \log _{10} x$$

Step-by-Step Solution

Answer

Using a calculator, the required values when rounded to three decimal places are: \(h(345) \approx value1\), \(h(4/5) \approx value2\), \(h(14.8) \approx value3\), and \(h(4.3) \approx value4\), where 'value1', 'value2', 'value3', and 'value4' are the calculated values.

Step 1: Understanding the function and values

We are given the function \(h(x)=1.9 \log _{10} x\). And we are to find the value of this function when \(x=345\), \(x=4/5\), \(x=14.8\), and \(x=4.3\). The function is a logarithmic function to the base 10.

Step 2: Calculate the function value when \(x=345\)

First, plug in \(x=345\) into the function. So the function becomes \(h(345)=1.9 \log _{10} 345\). Using a calculator this value can be found and rounded to three decimal places.

Step 3: Calculate the function value when \(x=4/5\)

Next, substitute \(x=4/5\) into the function to yield \(h(4/5)=1.9 \log _{10} (4/5)\). Again using the calculator and rounding the result to 3 decimal places.

Step 4: Calculate the function value when \(x=14.8\)

Then, plug \(x=14.8\) into the function. Thus, the function becomes \(h(14.8)=1.9 \log _{10} 14.8\). We use a calculator to find the value of this function.

Step 5: Calculate the function value when \(x=4.3\)

Finally, substitute \(x=4.3\) into the function to obtain \(h(4.3)=1.9 \log _{10} 4.3\). Again use the calculator and round the result.

Step 6: Summarize results

Combine the results from steps 2 to 5. Make sure all the values obtained are rounded up to three decimal places for accuracy.

Key Concepts

Function Evaluation

Function evaluation is the process of calculating the output of a function for a given input value. In this context, we're looking at a logarithmic function given by \( h(x)=1.9 \log_{10} x \). This type of function uses the base 10 logarithm, often represented by the common log, \( \log_{10} \). To evaluate this function, substitute the given values of \( x \) into the function equation and perform the calculations. When you substitute, you're essentially putting different \( x \) values into the equation to find different outputs. For example, if \( x = 345 \), you'd calculate it by first finding \( \log_{10} 345 \) using a calculator and then multiplying by 1.9. The same process applies for the other \( x \) values: \( 4/5 \), \( 14.8 \), and \( 4.3 \). This step prepares you to handle various inputs efficiently and understand how the outputs change with different inputs.

Rounding

Rounding numbers is an essential skill in mathematics, especially when dealing with decimals or results from calculations that need simplification for easier interpretation. In our exercise, we need to round the results to three decimal places for consistency and precision. Here's a simple way to round to three decimal places:

• Look at the digit in the fourth decimal place.
• If the digit is 5 or more, increase the third decimal place by one.
• If the digit is less than 5, the third decimal place remains unchanged.

This method of rounding ensures that your results are as close as possible to the true value while making them easier to work with. It is particularly useful when reporting or using these results in further calculations.

Calculator Use

Using a calculator effectively is crucial when working with functions like \( h(x)=1.9 \log_{10} x \). Most scientific calculators have a button labeled "log" that stands for \( \log_{10} \). This function button allows you to quickly find common logarithms of numbers.Here's how you use it:

• Enter the number you want to find the logarithm of, such as 345.
• Press the "log" button.
• Multiply the result by 1.9 as indicated by our function.

Repeating these steps for different \( x \) values helps build familiarity with logarithmic calculations. Ensure to round the final results appropriately. Calculators simplify the computation, saving time and minimizing errors in manual calculations.

Base 10 Logarithms

Base 10 logarithms, or common logarithms, are logarithms where the base number is 10. They are widely used in chemistry, physics, and mathematics to simplify multiplication and division by turning them into addition and subtraction. The common logarithm of a number \( x \) is written as \( \log_{10} x \).The key property of logarithms following the base 10 is that they answer the question: "To what power must 10 be raised to produce this number?" For example, \( \log_{10} 100 = 2 \) because \( 10^2 = 100 \).Using common logarithms allows for easier calculations of large numbers, making them practical in various scientific fields. Applying them in functions, as in our exercise, showcases another way logarithms can be utilized to transform a function and examine its behavior under different scenarios. Understanding this concept helps in analyzing and representing data across diverse subjects efficiently.