Problem 89 - College Algebra with Modeling and Visualization

Question

Solve each equation. Approximate answers to four decimal places when appropriate. $$4 \ln x=3$$

Step-by-Step Solution

Answer

The solution for \(x\) is approximately 2.1170.

Step 1: Simplify the equation

The equation we have is \(4 \ln x = 3\). To simplify, we want to isolate \(\ln x\). Divide both sides by 4: \(\ln x = \frac{3}{4}\).

Step 2: Solve for x using the exponential function

To solve for \(x\), we need to express \(x\) in terms of \(e\), the base of natural logarithms. Recall the property \(y = \ln x\) means \(x = e^y\). So, we have \(x = e^{\frac{3}{4}}\).

Step 3: Compute the value of x

Now calculate the value of \(x\) by substituting \(\frac{3}{4}\) into the exponent in the equation \(x = e^{\frac{3}{4}}\). Using a calculator, evaluate the expression \(e^{0.75}\).

Step 4: Round the answer

The calculated value of \(x\) should be rounded to four decimal places. From a calculator, \(e^{0.75} \approx 2.1170\), so the approximate value of \(x\) is 2.1170.

Key Concepts

Natural Logarithms

Natural logarithms are a special kind of logarithm which has the base 'e'. This is not just any number but approximately 2.71828, an irrational and transcendental number often used in mathematics, especially in calculus.
In our equation, \(4 \ln x = 3\), the 'ln' stands for a natural logarithm. Whenever you see 'ln', you can understand it as asking "to what power should 'e' be raised to get x?" This is a very handy way to solve exponential growth problems or to simplify them.
Key points about natural logarithms:

• The base is always 'e'.
• It helps to convert multiplication into addition, making complex equations simpler.
• 'ln' shifts an exponential equation into a form that is easier to work with, typically in solving for x.

Understanding these points makes natural logs a useful tool for solving various mathematical equations, especially when dealing with exponential functions.

Exponential Functions

Exponential functions involve expressions where a constant base is raised to a variable exponent. When working with natural logarithms, this base is the number 'e'. In our exercise, we used the property \(\ln x = y\), which implies \(x = e^y\).
This is the reverse operation of taking a logarithm and is crucial for solving equations where the variable is inside the logarithm. The key to unlocking these equations is understanding how to use exponentiation effectively.
Important things about exponential functions:

• The function grows quickly; this is due to the nature of the exponent.
• Used in modeling real-world scenarios like population growth, radioactive decay, and interest calculations.
• The inverse of an exponential function is a logarithmic function.

When you encounter exponential functions in equations, you can often simplify or solve them using the property that switches them back to logarithms, and vice versa. This interplay between exponentials and logarithms is a central theme in algebra.

Rounding Decimals

Rounding is the process of adjusting the numbers to make them easier to work with, especially when you need an approximate value. In mathematics, when dealing with complex numbers, a precise calculation might not be necessary.
For the equation in question, the result after solving the exponential function \(e^{0.75}\) is approximately 2.1170. This result is rounded to four decimal places. Rounding helps to share a comprehensible answer with fewer digits while maintaining reasonable accuracy.
Consider these steps when rounding:

• Identify how many decimal places you need (in this case, four).
• If the next digit beyond your last desired decimal place is 5 or more, round up the last digit. If it is 4 or less, keep it the same.
• Express the number with the adjustments made.

Rounding is used frequently in mathematics and helps especially in cases where calculations are performed by hand or when an exact answer is not required.