Problem 59 - Precalculus with Limits

Question

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. \(g(x)=8 \ln x \quad x=0.05\)

Step-by-Step Solution

Answer

The value of the function \(g(x)=8 \ln x\) at \(x=0.05\) rounded to three decimal places is approximately -7.130.

Step 1: Substitute the value of \(x\)

Take the function \(g(x)\), and substitute \(x\) with \(0.05\). That would make the function as \(g(0.05)=8 \ln 0.05\).

Step 2: Evaluate the logarithm

Now we will compute the value of the logarithm. In order to do that, a calculator can be used to get the numerical value. The calculator should have a button named 'ln', which stands for natural logarithm. After typing in '0.05' and pressing the 'ln' button, multiply the result by 8.

Step 3: Round the result

The result given by the calculator needs to be rounded to three decimal places. To do this, look at the fourth decimal place. If it is 5 or higher, round up the third decimal place by 1. If it is lower than 5, leave the third decimal as is.

Key Concepts

Understanding Natural Logarithms

Natural logarithms are a special type of logarithm that use the base \(e\), where \(e\) is approximately 2.71828. The natural logarithm is denoted as \(\ln\), and it's very common in various mathematical and scientific computations.

The expression \(\ln x\) tells us how many times we multiply \(e\) to get a number \(x\). For example, \(\ln(1) = 0\) because \(e^0 = 1\). When you see \(\ln x\), think about finding the power to which \(e\) must be raised to obtain \(x\).

• Important in calculus and exponential growth models.
• Helps in solving equations involving exponential functions.
• Key in transforming multiplicative processes to additive processes.

Understanding \(\ln x\) will make it easier to evaluate and manipulate functions like \(g(x)=8\ln x\).

Using a Calculator for Logs

Working with natural logarithms often requires the use of a calculator. Most scientific calculators have a dedicated 'ln' button. Here's how to use it:

• First, make sure your calculator is in the correct mode, usually standard or scientific.
• Type the value you need the logarithm for, in this case, 0.05.
• Press the 'ln' button, and the calculator will display the natural logarithm of that number.
• Finally, multiply the result by any required coefficients, which in this exercise is 8.

Using these steps helps in calculating \(\ln(0.05)\) accurately. Practicing with the calculator improves speed and accuracy.

Rounding Decimals Properly

Rounding decimals is an essential skill, especially when dealing with calculations that need to be precise. Here's the step-by-step on how to round to three decimal places:

• Look at the fourth decimal place.
• If it's 5 or higher, increase the third decimal place by one.
• If it's lower than 5, keep the third decimal place unchanged.

For example, if the calculator shows \(-59.7221506\) after multiplying, you would keep the first three decimals, \(-59.722\). In our exercise, be precise to ensure accuracy in mathematical results. This practice is key to maintaining consistency and clarity across computational tasks.