Problem 15 - Applied Calculus

Question

Solve for \(t\) using natural logarithms. $$5 e^{3 t}=8 e^{2 t}$$

Step-by-Step Solution

Answer

\(t = \ln\left(\frac{8}{5}\right)\)

Step 1: Divide Both Sides

To isolate the exponential terms, start by dividing both sides of the equation by \(e^{2t}\): \[\frac{5 e^{3t}}{e^{2t}} = \frac{8 e^{2t}}{e^{2t}}\]This simplifies to:\[5 e^{t} = 8\]

Step 2: Isolate the Exponential Term

Next, divide both sides of the simplified equation by 5 to further isolate the exponential term:\[e^{t} = \frac{8}{5}\]

Step 3: Apply Natural Logarithms

To solve for \(t\), apply the natural logarithm (ln) to both sides of the equation:\[\ln(e^{t}) = \ln\left(\frac{8}{5}\right)\]This simplifies to:\[t = \ln\left(\frac{8}{5}\right)\]

Step 4: Express the Solution

Finally, express the solution for \(t\):\[t = \ln\left(\frac{8}{5}\right)\]

Key Concepts

Natural Logarithms

Natural logarithms are an important tool in mathematics, especially when dealing with exponential equations. The natural logarithm, often denoted by \( \ln(x) \), is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. Natural logarithms are particularly useful because they often appear in calculus and mathematical models of natural growth or decay.
- The inverse of the natural exponential function \( e^x \) is \( \ln(x) \). This means that \( \ln(e^x) = x \) for any real number \( x \).
- When you take the natural logarithm of an exponential expression, like \( e^t \), it simplifies directly to \( t \) since \( \ln(e^t) = t \).
- Natural logarithms help in transforming multiplicative processes into additive ones, which simplifies solving equations that involve exponential terms.
For the given problem, applying the natural logarithm to both sides of the equation after isolating \( e^t \) simplifies our task of finding \( t \) effectively.

Exponential Functions

Exponential functions have the form \( f(x) = a e^{bx} \), where \( a \) and \( b \) are constants, and \( e \) is Euler's number. These functions represent processes that grow or decay at a rate proportional to their current value, such as population growth, radioactive decay, or interest compounding.
- The key characteristic of exponential functions is that they have a constant percentage rate of change, unlike linear functions which have a constant absolute rate of change.
- Exponential growth occurs when the coefficient \( b \) is positive, causing the function to rise sharply. Exponential decay happens when \( b \) is negative, causing the function to decrease rapidly.
In our exercise, \( e^{3t} \) and \( e^{2t} \) are both exponential expressions. By dividing one exponential expression by the other, we are able to simplify the equation to a form where the unknown variable \( t \) can be isolated and solved.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. When dealing with exponential equations like \( 5 e^{3t}=8 e^{2t} \), we use various algebraic strategies to isolate the variable and solve for it.
- Begin by simplifying the equation, trying to get the variable by itself. Often, this involves dividing both sides by a term or expression to simplify further.
- After isolating the exponential term, use logarithms to solve for the variable within the exponent. This transforms the exponential equation into a linear form that's much easier to solve.
- In the original step-by-step solution, dividing both sides by \( e^{2t} \) reduces the powers, making it possible to isolate \( e^t \).
By applying natural logarithms, we transform the problem into a log equation, \( t = \ln\left( \frac{8}{5} \right) \), directly solving for \( t \). These strategies apply not only to scientific problems but also to everyday issues like calculating investments or analyzing population dynamics.