Problem 52 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use logarithms to solve the equation for \(t\). $$4 e^{t-1}=4$$

Step-by-Step Solution

Answer

The short answer to the question is: To solve the equation \(4e^{t-1} = 4\) for \(t\), first simplify the equation by dividing both sides by 4, resulting in \(e^{t-1} = 1\). Then, take the natural logarithm of both sides, yielding \((t-1) \ln(e) = \ln(1)\). Using logarithmic properties, simplify the equation to \((t-1)(1) = 0\), and, therefore, the solution is \(t = 1\).

Step 1: Simplify the equation

To simplify the equation, we should isolate the exponential term, which is \(e^{t-1}\). To do that, divide both sides of the equation by 4: $$e^{t-1} = 1$$

Step 2: Apply natural logarithm to both sides

Now, we will apply the natural logarithm (denoted as \(\ln\)) to both sides of the equation: $$\ln\left(e^{t-1}\right) = \ln(1)$$

Step 3: Use the logarithmic property

Recall the logarithmic property that allows us to move the exponent to the front as a factor: $$\ln(a^b) = b\ln(a)$$ Using this property, we rewrite the equation as: $$(t-1)\ln(e) = \ln(1)$$

Step 4: Simplify using known logarithm values

Since \(\ln(e)\) is equal to 1 and \(\ln(1)\) is equal to 0, our equation becomes: $$(t-1)(1) = 0$$

Step 5: Solve for \(t\)

Finally, we can solve for \(t\) by simply adding 1 to both sides: $$t = 1$$ Therefore, the solution for the given equation is \(t = 1\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. The general form is \(a^x\), where \(a\) is a constant, and \(x\) is the exponent. In the context of this exercise, we're dealing with the natural exponential function \(e^x\), where \(e\) is approximately equal to 2.718. This function is particularly important in calculus and real-world applications because it grows continuously and is the basis for the natural logarithm.

Exponential functions have unique properties:

• The function \(e^x\) is one-to-one and always positive, meaning it never touches the x-axis but gets infinitely closer to it as \(x\) approaches negative infinity.
• As \(x\) increases, \(e^x\) increases rapidly, exhibiting exponential growth.

In our exercise, we started with \(4e^{t-1} = 4\). By isolating \(e^{t-1}\), we transformed it into an equation more suitable for applying logarithms.

Natural Logarithms

The natural logarithm, denoted as \(\ln\), is the inverse of the natural exponential function \(e^x\). It answers the question, "To what power must \(e\) be raised to obtain a given number?" One of the crucial properties of natural logarithms is that \(\ln(e^x) = x\), making them particularly useful for solving equations involving exponential functions.

In the exercise, we applied the natural logarithm to both sides of the equation \(e^{t-1} = 1\):

• This operation helps us bring down the exponent \(t-1\), which allows us to solve for \(t\) using the property \(\ln(e^x) = x\).
• It's important to recall specific values: \(\ln(e) = 1\) and \(\ln(1) = 0\). These simplify calculations significantly.

Using these properties of natural logarithms, we broke the problem into manageable parts, leading us to the solution.

Solving Equations

Solving equations is a fundamental skill in mathematics, especially for equations involving exponential and logarithmic functions. The key steps usually involve simplifying the equation, applying an appropriate transformation, and solving for the unknown variable. In this exercise, we used logarithms to turn an exponential equation into a linear one.

Here’s a step-by-step process:

• First, divide or simplify the equation to isolate the exponential expression, making it easier to manage.
• Second, apply the natural logarithm, \(\ln\), which helps in simplifying the equation by leveraging properties of logarithms like \(\ln(e^x) = x\).
• Next, use specific logarithmic values that simplify the equation, such as \(\ln(1) = 0\) and \(\ln(e) = 1\).
• Finally, solve the simplified equation using basic algebraic techniques.

Through understanding this process, one can solve not just exponential equations but other equations that can be simplified using logarithms.