Problem 362 - Precalculus

Question

Solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate \(x\) to 3 decimal places. \(50 e^{-0.12 t}=10\) using the natural log

Step-by-Step Solution

Answer

The approximate solution is \( t \approx 13.412 \).

Step 1: Isolate the Exponential Expression

Start by isolating the exponential part of the equation. The original equation is:\[ 50 e^{-0.12t} = 10 \]Divide both sides by 50 to isolate the exponential term:\[ e^{-0.12t} = \frac{10}{50} \]This simplifies to:\[ e^{-0.12t} = 0.2 \]

Step 2: Apply the Natural Logarithm

Take the natural logarithm (ln) of both sides to eliminate the exponential expression:\[ \ln(e^{-0.12t}) = \ln(0.2) \]By the property of logarithms, \( \ln(e^x) = x \), this simplifies to:\[ -0.12t = \ln(0.2) \]

Step 3: Solve for t

To solve for \( t \), divide both sides by \(-0.12\):\[ t = \frac{\ln(0.2)}{-0.12} \]Using a calculator to find \( \ln(0.2) \):\[ \ln(0.2) \approx -1.6094 \]Now substitute this value:\[ t = \frac{-1.6094}{-0.12} \approx 13.412 \]

Key Concepts

Natural Logarithm

The natural logarithm, often represented as "ln," is a logarithm to the base of Euler's number, "e." Aprilides, it's frequently used in calculus and mathematical modeling. Euler's number, approximately equal to 2.71828, is an irrational number and forms the basis of the natural exponential function. It's important in continuous growth models, such as compound interest and population growth models. When you see an expression like \( \ln(e^x) = x \)\, it is a key property that simplifies calculations. Here, the natural logarithm and the exponential function are inverses of each other. This property is essential when solving exponential equations. For exponential equations containing \( e \), converting them with the ln function helps bring down the exponent, simplifying the process of finding the value of unknown variables.Remember, using the natural logarithm aligns with the properties of "e," helping make intricate calculations more manageable.

Logarithmic Properties

Logarithmic properties are hugely helpful when dealing with exponential equations. They let us transform an exponential expression into a simpler, solvable equation. Here are some core properties:

• Inverse Nature: The logarithm of a power, like \( \ln(e^x) \), equals the exponent \( x \). This inverse relationship allows easy simplification.
• Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \) enables breaking down products into sums.
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \) facilitates working with division.
• Power Rule: \( \log_b(M^n) = n \cdot \log_b(M) \) means the exponent can be moved as a multiplier.

In our provided solution, we utilized the inverse property, simplifying \( \ln(e^{-0.12t}) \) directly into \( -0.12t \). These logarithmic manipulations transform complex expressions into simpler algebraic equations, making solutions more accessible to grasp and calculate.

Calculator Approximations

Calculator approximations are crucial when solving equations that involve logarithms. They provide accurate, step-by-step values for calculations, especially for irrational numbers or complex expressions.When working through exponential equations, calculators help with:

• Calculating Logarithms: Functions compute ln (logarithm base \( e \)) precisely, as manual calculations of values like \( \ln(0.2) \) are complex.
• Decimal Approximation: Calculators approximate irrational numbers to specific decimal places, useful for equations depending on precision, like financial models.
• Simplifying Solutions: Aside from logarithms, performing multiplications, divisions, or exponentiations is quick and precise.

In the original example, after taking the logarithm, the solution required the approximation of \( \ln(0.2) \). With a calculator, this value \( \approx -1.6094 \) permitted the continuation of the problem-solving process, resulting in a precise approximation of \( t \approx 13.412 \). This illustrates how calculators streamline solving complex numerical problems, reducing human errors and ensuring clarity in results.