Problem 23 - Calculus

Question

Solve for \(t .\) Assume \(a\) and \(b\) are positive constants and \(k\) is nonzero. $$a=b e^{t}$$

Step-by-Step Solution

Answer

The solution for \(t\) is \(t = \ln \left(\frac{a}{b}\right)\).

Step 1: Isolate the Exponential Term

To solve for \(t\), start by isolating the exponential term \(e^{t}\). Divide both sides of the equation by \(b\) to achieve this. The equation becomes:\[\frac{a}{b} = e^{t}.\]

Step 2: Apply the Natural Logarithm

To solve for \(t\), take the natural logarithm (ln) of both sides. The natural logarithm is the inverse function of the exponential function, so applying it will bring \(t\) down from the exponent. The equation thus transforms to:\[\ln \left(\frac{a}{b}\right) = \ln(e^{t}).\]This simplifies to:\[\ln \left(\frac{a}{b}\right) = t.\]

Step 3: Conclude the Solution

We have isolated \(t\), and the solution to the equation is:\[t = \ln \left(\frac{a}{b}\right).\]

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a specific logarithmic function with the base \(e\), where \(e\) (approximately 2.718) is a mathematical constant known as Euler's number. It is widely used in calculus and exponential equations. The natural logarithm converts exponential growth processes into a linear form, making them easier to analyze.
In the context of solving exponential equations, the natural logarithm acts as the inverse of the exponential function. This means that if we have an equation like \( e^t \), taking the natural logarithm of both sides will help resolve the exponent, effectively "bringing down" the \(t\) from the exponent.
This property is essential when we need to solve for variables like \(t\) that appear as exponents in equations. For instance, applying \( \ln \) to both sides of the equation \( e^t = c \) results in \( t = \ln(c) \). This simplification is crucial for finding solutions easily.

Isolating Variables

When solving equations, isolating the variable of interest is a key step. This process involves manipulating the equation to get the variable alone on one side of the equation. This might be done by using arithmetic operations such as addition, subtraction, multiplication, or division.
In the given problem, you were asked to solve for \(t\) in the equation \( a = be^t \). The first step in solving this is to isolate the exponential term \( e^t \). We do this by dividing both sides by \( b \), resulting in \( \frac{a}{b} = e^t \).
By isolating \( e^t \), we make it possible to apply the natural logarithm and solve for \(t\). This methodical approach of getting one term by itself can be applied to various kinds of equations, facilitating the solution process.

Inverse Functions

Understanding inverse functions is a fundamental aspect of solving equations, especially those involving exponentials. An inverse function essentially reverses the operations of the original function. For exponential functions, the inverse is the logarithm.
In the original problem, the exponential function is \( e^t \), and its inverse is the natural logarithm. When applied to both sides of the equation \( \frac{a}{b} = e^t \), the natural logarithm "undoes" the exponential effect on \(t\). Therefore, \( \ln(e^t) = t \) because the natural log and exponential operations are inverses and effectively cancel each other out.
This use of inverse functions simplifies the problem and allows you to express \(t\) in terms of other known quantities, specifically \( \ln(\frac{a}{b}) \). Mastering this technique is very useful for solving various types of mathematical equations.