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Chapter 3: Problem 56

Use logarithms to solve the equation for \(t\). $$\frac{200}{1+3 e^{-0.3 t}}=100$$

Short Answer

Expert verified
The short version of the answer is: To solve the equation \(\frac{200}{1+3 e^{-0.3 t}}=100\) for \(t\), first simplify the equation to get \(\frac{1}{3} = e^{-0.3t}\). Next, take the natural logarithm of both sides: \(\ln{\frac{1}{3}} = -0.3t\). Finally, solve for \(t\): \(t \approx \frac{\ln{\frac{1}{3}}}{-0.3} \approx 3.66\).

Step by step solution

01

Simplify the equation

First, we need to simplify the given equation: \(\frac{200}{1+3 e^{-0.3 t}}=100\) To do this, we can first multiple both sides by \((1+3e^{-0.3t})\): \(200 = 100(1+3e^{-0.3t})\) Now, divide both sides by 100: \(2 = 1+3e^{-0.3t}\) And then, subtract 1 from both sides to isolate the exponential term: \(1 = 3e^{-0.3t}\) Finally, divide by 3 to isolate the exponential term completely: \(\frac{1}{3} = e^{-0.3t}\)
02

Take the natural logarithm of both sides

Now, apply the natural logarithm to both sides of the equation: \(\ln{\frac{1}{3}} = \ln{e^{-0.3t}}\)
03

Use properties of logarithms

Using the property of logarithm, where \(\ln{a^b} = b\cdot\ln{a}\), we can simplify the right-hand side of the equation: \(\ln{\frac{1}{3}} = -0.3t\cdot\ln{e}\) Since \(\ln{e} = 1\), this now becomes: \(\ln{\frac{1}{3}} = -0.3t\)
04

Solve for \(t\)

To solve for \(t\), we need to divide both sides of the equation by '-0.3': \(t = \frac{\ln{\frac{1}{3}}}{-0.3}\) So, \(t \approx \frac{-1.098}{-0.3}\) And, \(t \approx 3.66\) Thus, the solution for the equation is \(t \approx 3.66\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. These functions are crucial in modeling numerous real-world phenomena, such as population growth, radioactive decay, and certain financial markets.
When solving equations involving exponential functions, it鈥檚 essential to identify and isolate the exponential component as the first step. In our original exercise, the exponential function is represented by the expression \(e^{-0.3t}\).
Understanding this concept helps us manipulate the equation effectively by isolating the exponential term, which makes it easier to apply logarithmic operations in subsequent steps.
Natural Logarithms
Natural logarithms use "\(e\)" as their base and are denoted by \(\ln\). They are pivotal when working with exponential equations because of the special relationship between \(e\) and its logarithm, which simplifies calculations significantly.
In the context of our problem, the natural logarithm is applied to both sides of the equation to eliminate the base \(e\) and linearize the expression. The step \(\ln{e^{-0.3t}}\) simplifies to \(-0.3t\cdot\ln{e}\), leveraging the property \(\ln{e}=1\).
This simplification is crucial as it transforms an otherwise complex exponential equation into a simpler linear form that is much easier to solve.
Equation Solving
Equation solving involves finding the value of the variable that satisfies the equation. In our original equation, solving for \(t\) requires several algebraic manipulations and the application of logarithmic properties.
The process typically involves:
  • Isolating the variable component on one side of the equation.
  • Applying properties of logarithms or exponents appropriately.
  • Simplifying the equation systematically through operations such as addition, subtraction, multiplication, or division.
By transforming the exponential equation into a linear form using logarithms, we facilitate simpler and more straightforward mathematical operations to find \(t\).
This makes equation solving approachable by reducing the complexity of the functional forms involved.
Mathematical Simplification
Mathematical simplification involves strategically reducing an expression to its most basic and manageable form. It requires knowing operational tactics and properties of the elements involved, such as logarithms or exponents, to make calculations easier.
During simplification in our exercise, we manipulate the equation using basic algebraic techniques:
  • Multiplying to eliminate fractions.
  • Isolating terms to manage multiplicative components.
  • Utilizing logarithmic properties to convert complex structures into simple linear ones.
Ultimately, simplification ensures clarity and ease in solving equations by systematically eliminating complexity and focusing directly on the elements that need solving.
This approach not only helps in solving the present equation but also builds a strong foundational understanding for tackling varied mathematical problems.

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Most popular questions from this chapter

The world population at the beginning of 1990 was \(5.3\) billion. Assume that the population continues to grow at the rate of approximately \(2 \% / y e a r\) and find the function \(Q(t)\) that expresses the world population (in billions) as a function of time \(t\) (in years), with \(t=0\) corresponding to the beginning of 1990 . Using this function, complete the following table of values and sketch the graph of the function \(Q\).

Express each equation in logarithmic form. $$32^{3 / 5}=8$$

Express each equation in logarithmic form. $$10^{-3}=0.001$$

Use the laws of logarithms to solve the equation. $$\log _{2}(2 x+5)=3$$

Use logarithms to solve the equation for \(t\). $$\frac{50}{1+4 e^{0.2 t}}=20$$
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