Problem 21 - Contemporary Precalculus

Question

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$2.1 e^{(x / 2) \ln 3}=5$$

Step-by-Step Solution

Answer

Answer: The approximate value of x is 1.482.

Step 1: Isolate the term containing x

Divide both sides of the equation by 2.1 to isolate the term containing \(x\): $$e^{(x / 2) \ln 3}=\frac{5}{2.1}$$

Step 2: Apply logarithm properties

Take the natural logarithm of both sides to eliminate the exponential function: $$(x / 2) \ln 3=\ln\left(\frac{5}{2.1}\right)$$

Step 3: Express the solution in terms of natural logarithms

Multiply both sides of the equation by \(2\) and then divide by \(\ln 3\) to isolate \(x\): $$x=\frac{2\ln\left(\frac{5}{2.1}\right)}{\ln 3}$$

Step 4: Use a calculator to find an approximation for the answer

Using a calculator, we get: $$x\approx\frac{2\ln\left(\frac{5}{2.1}\right)}{\ln 3}\approx 1.482$$ So, the solution is approximately \(x\approx 1.482\).

Key Concepts

Natural Logarithms

Natural logarithms are logarithms with base \(e\), where \(e\) is a constant approximately equal to 2.71828. They are denoted as \(\ln\) and are fundamental in calculus and various scientific calculations. Natural logarithms have interesting properties that simplify the manipulation and solving of equations, especially those involving exponential functions.
Using natural logarithms, like in our exercise, helps to handle equations where the unknown variable appears in the exponent. In this specific problem, taking the natural logarithm of both sides allows us to "bring down" the exponent for easier manipulation. This is crucial for isolating the variable and solving for it.

Isolation of Variables

Isolation of variables is a key step in solving equations. It refers to the process of rearranging the equation so that the variable in question stands alone on one side. This was the first step in our exercise, where we divided both sides of the equation by 2.1 to isolate the exponential term containing \(x\).

This step is critical as it sets the stage for further transformation and simplification of the equation. By clearly separating the term you are solving for, you make it easier to use other techniques, such as applying logarithms, to find a solution.

• Identify terms with the variable.
• Perform operations to isolate these terms.
• Simplify the equation to solve for the variable efficiently.

Approximation Techniques

Approximation techniques are useful when an exact symbolic solution is difficult or impossible to obtain. Once the variable was isolated and expressed in terms of logarithms, a calculator was used to approximate the exact value of \(x\).

In practical applications, especially where precision to many decimal places is unnecessary or not possible, approximation provides a sufficient solution. Calculators or computational software tools often carry out the approximation for expressions involving logarithms or other complex functions.

• Use calculators for functions not easily simplified.
• Rely on approximation for real-world applications needing speed over exactness.
• Understand precision requirements to choose appropriate techniques.

Logarithmic Properties

Logarithmic properties are rules that simplify equations involving logarithms, making them a powerful tool for solving exponential equations. In this exercise, one key property was utilized: the ability to take the logarithm of both sides of an equation to remove the exponential function from the variable.
These properties include:

• \(\ln(ab) = \ln a + \ln b\)
• \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\)
• \(\ln(a^b) = b\ln a\)

In our example, after isolating the exponential term, applying the natural logarithm to both sides allowed us to transform the problem from an exponential equation into an algebraic equation. This eventually enabled us to solve for \(x\) efficiently.