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Chapter 4: Problem 105

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln \sqrt{x+2}=1$$

Short Answer

Expert verified
Based on the steps above, the solution to the equation is \(x = 0.718\)

Step by step solution

01

Square both sides to eliminate the square root

Firstly, the logarithm is taking the square root of \(x+2\). To remove the square root, square both sides of the equation. This gives \((\ln\sqrt{x+2})^2=1^2\), which simplifies further to \(\ln^2 (x+2)=1\).
02

Transform the log equation into exponential form

Logarithmic and exponential functions are inverses of each other. Therefore, to simplify the equation, it is helpful to write the equation in exponential form. The base of the natural log, \(\ln\), is \(e\), thus \(e^1 = x+2\). This simplifies to \(e = x +2\).
03

Solve for x

To isolate x, subtract 2 from both sides. This gives \(x = e - 2 \).
04

Approximate to three decimal places

Since e is approximately equal to 2.718, the final solution will be \(x = 2.718 - 2\), that is \(x = 0.718\). However, because the problem asks for a solution rounded to three decimal places, adjust this to \(x = 0.718\).

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Key Concepts

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

Understanding the Natural Logarithm
The natural logarithm, often denoted as \(\ln\), is a special kind of logarithm. It has the base \(e\), where \(e\) is an irrational number approximately equal to 2.718. Natural logarithms are used primarily in calculus and complex calculations because they simplify many derivative and integration problems.

When we write \(\ln(x)\), we are essentially asking, "To what power must \(e\) be raised to achieve \(x\)?" For instance, \(\ln(e) = 1\) because \(e^1 = e\).

  • This concept is pivotal in solving equations that involve exponential growth or decay.
  • Understanding \(\ln\) helps in unraveling exponential functions.
  • It is particularly helpful for solving equations involving unknown exponents.
Natural logarithms are incredibly useful in modeling real-world phenomena such as population growth, radioactive decay, and financial calculations related to compound interest.
Exploring Exponential Functions
Exponential functions are mathematics' way of describing rapid growth or decay. In an exponential function, the variable appears in the exponent rather than in the base. For example, the function \(f(x) = e^x\) is an exponential function. Here, \(e\) is the base, and \(x\) is the exponent.

These functions have distinctive characteristics:
  • Their graphs are curved, not straight.
  • They always pass through the point \((0, 1)\) on a graph, as any number raised to the power of zero equals one.
  • The function \(e^x\) increases rapidly as \(x\) becomes larger, a pattern seen in processes like compound interest and population growth.
Because logarithmic and exponential functions are inverses, they are key to solving various types of equations. Understanding how to manipulate these functions can be incredibly powerful for students, enabling them to tackle a wide array of mathematical problems.
The Role of Approximations in Math
Approximations are essential in mathematics, especially when dealing with irrational numbers like \(e\). Since exact values of such numbers cannot be expressed finitely, we often use approximations like \(e \approx 2.718\) to make calculations manageable.

Here鈥檚 why approximations are valuable:
  • They simplify complex calculations.
  • They allow for practical conclusions and decisions, especially in real-world problems.
  • They help in achieving a close estimate when an exact number isn't essential.
However, it's crucial to note the contexts where precision matters, such as scientific calculations or financial forecasting. By understanding how to use approximations effectively, students can improve their skills in estimation and problem-solving. The exercise of rounding to three decimal places, as seen here, is a common application of practical approximation.
Inverse Functions and Their Significance
Inverse functions are operations that "undo" each other. For example, the inverse of addition is subtraction, and similarly, the inverse of a logarithmic function is the exponential function. In mathematical terms, if \(f(x)\) is a function and \(g(x)\) is its inverse, then \(f(g(x)) = x\) and \(g(f(x)) = x\).

In the context of natural logarithms and exponentials:
  • The function \(y = \ln(x)\) has the inverse \(x = e^y\).
  • This relationship is crucial when solving equations, as it allows for switching between forms to find solutions.
  • Inverse functions provide a dual perspective, enabling students to solve problems from different angles.
Mastering inverse functions enhances students' understanding of mathematical relationships, equipping them with versatile tools for both theoretical and practical applications.

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Most popular questions from this chapter

A beaker of liquid at an initial temperature of \(78^{\circ} \mathrm{C}\) is placed in a room at a constant temperature of \(21^{\circ} \mathrm{C}\) The temperature of the liquid is measured every 5 minutes for a period of \(\frac{1}{2}\) hour. The results are recorded in the table, where \(t\) is the time (in minutes) and \(T\) is the temperature (in degrees Celsius). $$\begin{array}{|c|c|}\hline \text { Time, } t 0 & \text { Temperature, } T\\\\\hline 0 & 78.0^{\circ} \\\5 & 66.0^{\circ} \\\10 & 57.5^{\circ} \\\15 & 51.2^{\circ} \\\20 & 46.3^{\circ} \\\25 & 42.5^{\circ} \\\30 & 39.6^{\circ} \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find a linear model for the data. Use the graphing utility to plot the data and graph the model in the same viewing window. Do the data appear linear? Explain. (b) Use the regression feature of the graphing utility to find a quadratic model for the data. Use the graphing utility to plot the data and graph the model in the same viewing window. Do the data appear quadratic? Even though the quadratic model appears to be a good fit, explain why it might not be a good model for predicting the temperature of the liquid when \(t=60.\) (c) The graph of the temperature of the room should be an asymptote of the graph of the model. Subtract the room temperature from each of the temperatures in the table. Use the regression feature of the graphing utility to find an exponential model for the revised data. Add the room temperature to this model. Use the graphing utility to plot the original data and graph the model in the same viewing window. (d) Explain why the procedure in part (c) was necessary for finding the exponential model.

(a) complete the table to find an interval containing the solution of the equation, (b) use a graphing utility to graph both sides of the equation to estimate the solution, and (c) solve the equation algebraically. Round your results to three decimal places. $$\ln 2 x=2.4$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & 2 & 3 & 4 & 5 & 6 \\\\\hline \ln 2 x & & & & & \\\\\hline\end{array}$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$2 x^{2} e^{2 x}+2 x e^{2 x}=0$$

Use a graphing utility to create a scatter plot of the data. Decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model. $$(1,5.0),(1.5,6.0),(2,6.4),(4,7.8),(6,8.6),(8,9.0)$$

Use the regression feature of a graphing utility to find a logarithmic model \(y=a+b \ln x\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(3,14.6),(6,11.0),(9,9.0),(12,7.6),(15,6.5)$$
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