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Chapter 4: Problem 51

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(2 \ln (4-3 t)+1=7\)

Short Answer

Expert verified
Exact solution: \(\left\{ \frac{4 - e^3}{3} \right\}\), Approximate solution: \(-5.3618\)\

Step by step solution

01

- Isolate the logarithmic term

Subtract 1 from both sides of the equation to isolate the logarithmic term: yielding: o_minus1_equation: \[2 \ln(4 - 3t) = 6\]
02

- Divide both sides by 2

Divide both sides of the equation by 2 to simplify: a_divide_equation: \[\ln(4 - 3t) = 3\].
03

- Exponentiate both sides

Exponentiate both sides using the base of the natural logarithm (Euler's number, e) to eliminate the logarithm: exponentiate: \[4 - 3t = e^3\].
04

- Solve for t

Isolate t by first subtracting 4 from both sides and then dividing by -3: yielding to: \[t = \frac{4 - e^3}{3}\]
05

- Calculate approximate solution

Approximate the numerical value of the solution to 4 decimal places using a calculator: \(e^3 \approx 20.0855\)Approximated equation: \[t \approx \frac{4 - 20.0855}{3} = \frac{-16.0855}{3} \approx -5.3618\].
06

- Write the solution set

The solution set for the equation, both exact and approximate, is:\[\left\{ \frac{4 - e^3}{3} \right\}\] for the exact solution and \[-5.3618\] for the approximate solution.

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Key Concepts

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

Natural Logarithm
The natural logarithm, denoted as \(\text{ln}\), is a logarithm to the base \(e\). The constant \(e\) is approximately equal to 2.71828 and it is a fundamental mathematical constant, also known as Euler's number. The natural logarithm is the inverse function of the exponential function, meaning that if \(y = e^x\), then \(x = \text{ln}(y)\). In the given exercise, \( \text{ln}(4 - 3t) \) is the natural logarithmic term isolated and simplified.
Exact Solutions
An exact solution to a logarithmic equation provides the solution in a precise, non-rounded form. In the given exercise, the exact solution is found by isolating \(t\) in the equation \( 2 \text{ln}(4 - 3t) + 1 = 7 \). By subtracting 1, dividing by 2, exponentiating, and solving for \(t\), we obtain: \[ t = \frac{4 - e^3}{3} \]. This solution is exact because it is expressed using known mathematical constants without approximation.
Approximate Solutions
Approximate solutions are numerical values computed to a specified level of precision, usually to simplify calculations or interpretations. In our example, the approximate solution is derived by calculating \( e^3 \) and rounding the result. Since \( e^3 \) is approximately 20.0855, the value of \( t \) can be computed as: \[ t \approx \frac{4 - 20.0855}{3} \approx \frac{-16.0855}{3} \approx -5.3618 \]. This approximation helps make the result more accessible.
Exponentiation
Exponentiation is the process of raising a quantity to a power. In logarithmic equations, exponentiating both sides is often necessary to eliminate the logarithm. For the provided equation, we exponentiate both sides using the base \(e\) to transform \( \text{ln}(4 - 3t) = 3 \) into \( 4 - 3t = e^3 \). This step is crucial to convert the logarithmic form back into a linear form that can be solved algebraically.
College Algebra
College Algebra covers a variety of foundational topics, including logarithmic and exponential functions. Understanding how to solve logarithmic equations like the one provided here is a fundamental skill. It involves multiple algebraic techniques such as isolating variables, performing arithmetic operations, exponentiation, and understanding the properties of logarithms. Mastery in these areas bolsters mathematical proficiency essential for more advanced studies.

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Most popular questions from this chapter

The isotope of plutonium \({ }^{238} \mathrm{Pu}\) is used to make thermoelectric power sources for spacecraft. Suppose that a space probe is launched in 2012 with \(2.0 \mathrm{~kg}\) of \({ }^{238} \mathrm{Pu}\) a. If the half-life of \({ }^{238} \mathrm{Pu}\) is \(87.7 \mathrm{yr}\), write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the quantity \(Q(t)\) of \({ }^{238} \mathrm{Pu}\) left after \(t\) years. b. If \(1.6 \mathrm{~kg}\) of \({ }^{238} \mathrm{Pu}\) is required to power the spacecraft's data transmitter, for how long will scientists be able to receive data? Round to the nearest year.

Write the domain in interval notation. $$ d(x)=\log \left(\frac{1}{\sqrt{x+8}}\right) $$

The population of Canada \(P(t)\) (in millions) since January \(1,1900,\) can be approximated by $$P(t)=\frac{55.1}{1+9.6 e^{-0.02515 t}}$$ where \(t\) is the number of years since January 1,1900 . a. Evaluate \(P(0)\) and interpret its meaning in the context of this problem. b. Use the function to predict the Canadian population on January \(1,2015 .\) Round to the nearest million. c. Use the function to predict the Canadian population on January 1,2040 . d. Determine the year during which the Canadian population will reach 45 million. e. What value will the term \(\frac{9.6}{e^{0.02515 t}}\) approach as \(t \rightarrow \infty\) ? f. Determine the limiting value of \(P(t)\).

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\ln x+\ln (x-4)=\ln (3 x-10)\)

a. The populations of two countries are given for January 1,2000 , and for January 1,2010 . Write a function of the form \(P(t)=P_{0} e^{k t}\) to model each population \(P(t)\) (in millions) \(t\) years after January 1, 2000.$$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Population } \\ \text { in 2000 } \\ \text { (millions) } \end{array} & \begin{array}{c} \text { Population } \\ \text { in 2010 } \\ \text { (millions) } \end{array} & \boldsymbol{P}(t)=\boldsymbol{P}_{0} e^{k t} \\ \hline \text { Switzerland } & 7.3 & 7.8 & \\ \hline \text { Israel } & 6.7 & 7.7 & \\ \hline \end{array}$$ b. Use the models from part (a) to predict the population on January \(1,2020,\) for each country. Round to the nearest hundred thousand. c. Israel had fewer people than Switzerland in the year 2000 , yet from the result of part (b), Israel will have more people in the year \(2020 ?\) Why? d. Use the models from part (a) to predict the year during which each population will reach 10 million if this trend continues.
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