Problem 73 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

a. Given that \(2^{x}=e^{k x}\), find \(k\). b. Show that, in general, if \(b\) is a nonnegative real number, then any equation of the form \(y=b^{x}\) may be written in the form \(y=e^{k x}\), for some real number \(k\).

Step-by-Step Solution

Answer

In summary, for the given equation \(2^{x}=e^{k x}\), we found that \(k = \ln{2}\). Additionally, we showed that the general equation of the form \(y=b^{x}\) can be rewritten as \(y=e^{k x}\), where \(k = \ln{b}\).

Step 1: Rewrite the given equation using logarithms

To find the value of \(k\), we will first rewrite the given equation \(2^{x}=e^{k x}\) using logarithms. We can write this using the natural logarithm (ln) which has a base of 'e'. Therefore, we have: \[x \ln{2} = kx \ln{e}\]

Step 2: Solve for k

Now, we'll solve the equation from Step 1 for k: \[x \ln{2} = kx \ln{e}\] Since \(x\) is present on both sides, and \(\ln{e} = 1\), we can rewrite the equation as: \[k = \frac{\ln{2}}{\ln{e}} = \ln{2}\] Therefore, \(k = \ln{2}\). #b. Showing that the general form of y=b^x can be written as y=e^(kx)#

Step 1: Rewrite the equation using logarithms

Now, let's take the general form of the equation, \(y=b^{x}\), and rewrite it using logarithms, just like we did with the given equation. We can use the natural logarithm as well: \[y = b^{x} \Rightarrow \ln{y} = x \ln{b}\] Now, we want to rewrite this equation in the form \(y = e^{kx}\).

Step 2: Introduce exponentials

In order to rewrite the equation in the form \(y = e^{kx}\), we need to introduce exponentials. Recall that \(\ln{a^b} = b\ln{a}\) and \(\ln{e^b}=b\). So, we can rewrite the equation from step 1 as: \[y = e^{\ln{b^x}}\] Now, using the property mentioned above, we get: \[y = e^{x\ln{b}}\]

Step 3: Find the value of k

Now that we have the equation in the form \(y = e^{x\ln{b}}\), we can identify k as the coefficient of x in the exponent: \[k = \ln{b}\]

Step 4: Rewrite the equation in the desired form

Finally, substitute the value of k in the equation \(y = e^{kx}\): \[y = e^{\ln{b}x}\] Thus, we have shown that any equation of the form \(y=b^{x}\) can be written as \(y=e^{kx}\), for some real number \(k\).

Key Concepts

Logarithms

Logarithms are mathematical tools that help us solve equations involving exponentials by transforming multiplication into addition, division into subtraction, powers into products, and roots into division. They serve as the inverse operation to exponentiation. If given an expression like \(b^x\), you can use logarithms to "bring down the exponent" and solve for unknowns more easily.

If you have an equation like \(b^x = y\), taking the logarithm of both sides (with base \(b\)) gives:

• \(\log_b(y) = x\)

This property is powerful because it allows us to solve equations where the variable is an exponent. In many mathematical contexts, especially while dealing with growth processes (like population, radioactive decay), logarithms are essential. Logarithms make complex equations manageable and offer insights into the relationships between the quantities involved.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), has a special significance in mathematics. It uses the base \(e\), a mathematical constant approximately equal to 2.71828. The natural logarithm simplifies various calculus equations and often appears in formulas involving continuous growth or decay processes.

When you have a situation involving \(e^x\), using the natural logarithm is useful because:

• \(\ln(e^x) = x\)
• \(\ln(e) = 1\)

These properties make the natural logarithm a favorite tool for mathematicians because it elegantly simplifies expressions and serves as an efficient method to isolate variables. In exponential equations, reformatting an expression like \(e^{kx}\) using \(\ln\) makes it easier to find the value of \(k\), providing deeper insights into the behavior of exponential models.

Mathematical Equations

Mathematical equations are symbolic statements that express the equality between two expressions. They are foundational in mathematics for expressing and solving problems. The ability to manipulate equations is crucial for uncovering relationships between variables, particularly in solving for unknowns.

Equations involving exponential functions, like \(2^x = e^{kx}\), often require transformation using logarithms to be solved effectively. The equation \(2^x = e^{kx}\) can be approached by:

• Applying logarithmic functions to both sides to isolate terms.
• Using properties of logarithms to simplify complex expressions.

In the solution path, we demonstrate such transformations to find \(k\). This dynamic approach to equations allows for flexible manipulation, turning taxing problems into simpler tasks. By understanding these methodologies, you're better equipped to tackle a wide range of mathematical challenges beyond just textbook exercises.