Problem 16 - Applied Calculus

Question

For Problems \(1-16,\) 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

Start by dividing both sides of the equation by "\( e^{2t} \)" to simplify the exponent terms. This gives us:\[ \frac{5 e^{3t}}{e^{2t}} = 8 \] Simplifying the left side, we have: \[ 5 e^{t} = 8 \]

Step 2: Isolate the Exponential Function

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

Step 3: Apply Natural Logarithms

Apply the natural logarithm (ln) to both sides of the equation to eliminate the exponent: \[ \ln(e^{t}) = \ln\left(\frac{8}{5}\right) \] Using the logarithmic identity \( \ln(e^{t}) = t \), our equation simplifies to: \[ t = \ln\left(\frac{8}{5}\right) \]

Key Concepts

Understanding Natural Logarithms

Logarithms are used to solve equations involving exponential functions. A natural logarithm is a logarithm to the base of the number "e," where "e" is approximately 2.71828. Natural logarithms are often written using the abbreviation "ln." Using natural logarithms, we can solve equations involving the number "e" by bringing exponents down to a simpler, linear form.

In the given exercise, after isolating the exponential expression, we apply the natural logarithm to both sides. The core idea behind this is the identity:

• When you take the natural logarithm of an expression in the form of \(e^{x}\), it simplifies the equation because \(\ln(e^{x}) = x\). This transformation makes it easier to solve for the variable, in our case, "t."

By using natural logarithms, we have a powerful tool to convert complex exponential equations into manageable linear equations.

Exploring Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In our problem, the base is the number "e." When solving exponential equations, one common goal is to isolate the exponential term. This makes applying logarithms straightforward.

Here's how it works:

• An exponential function such as \(5e^{3t}=8e^{2t}\) means we are dealing with both terms that grow exponentially with "t."
• By dividing by \(e^{2t}\), we simplify our equation so that we are left with \(5e^{t}\) instead of \(5e^{3t}\).

Once isolated, the exponential component can easily be managed with logarithms, reducing the equation into more manageable computations.

Logarithmic Identities Simplified

Logarithmic identities are essential tools in manipulating logarithms in equations. These identities can simplify complex expressions, making it easier to isolate variables. Two key identities used in this problem include:

- The Identity \(\ln(e^{x}) = x\) is crucial because it simplifies the exponential term directly into the linear term. This specific identity essentially says, "if you take the natural log of an exponential function, you're left with what's in the exponent." - Another important identity involves the logarithm itself: \(\ln(a\cdot b) = \ln(a) + \ln(b)\). While not directly used in our solution, understanding this helps when dealing with products inside logarithms.Understanding these identities equips you with strategies to tackle logarithmic equations. By mastering these rules, even complex equations become straightforward to solve, as seen in solving for "t" in our example problem.