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Chapter 3: Problem 71

Solve each logarithmic equation in Exercises \(49-92\). Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log _{2}(x+2)-\log _{2}(x-5)=3$$

Short Answer

Expert verified
The exact answer is \(x = 6\)

Step by step solution

01

Apply log properties

Applying logarithmic properties to the equation, the difference of logs equals the log of the quotient. Therefore, we can say that \(\log _{2}\left(\frac{x+2}{x-5}\right) = 3\)
02

Use equal logs rule

The equal log rule states that if \(log_a(b) = c\), then \(a^c = b\). Therefore, applying this rule, we get \(\frac{x+2}{x-5} = 2^3 = 8\)
03

Simplify to form a quadratic equation

To solve this equation, multiply both sides of the equation by \(x-5\), so that we get \(x+2 = 8(x-5)\). Expanding the right side of the equation gives us \(x+2 = 8x - 40\). Bringing all terms to one side of the equation gives a quadratic equation: \(7x - 42 = 0\)
04

Solve the quadratic equation

Dividing through by 7, we find that \(x = 6\)
05

Validate the solution

Validating the value of \(x\), 6 has to be greater than 5 because of the function \(\log _{2}(x-5)\), else the output will be undefined. So, \(x = 6\) is a valid solution

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Key Concepts

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

Properties of Logarithms
Logarithms have unique properties that make them especially useful for solving equations. These properties allow us to manipulate and simplify expressions involving logs. One crucial property is the subtraction of logarithms rule:
  • Difference Rule: \ \(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \)
In the original exercise, this rule transforms the equation \(\log_{2}(x+2) - \log_{2}(x-5) = 3\) into \(\log_{2}\left(\frac{x+2}{x-5}\right) = 3\). This transformation simplifies the problem, making it easier to solve.
The properties of logarithms are essential as they help convert multiplication into addition, division into subtraction, and exponentiation into multiplication. These conversions bring forward simpler-to-handle mathematical expressions.
Domain of a Function
Understanding the domain of a function is crucial when dealing with logarithmic equations. The domain refers to all possible input values (\(x\)) for which a function is defined.
For logarithms, such as \(\log_{2}(x-5)\), the domain is constrained by the condition that whatever is inside the log must be greater than zero. Therefore:
  • Inside Log Greater Than Zero: For \(\log_{2}(x-2)\), the condition is \(x-5 > 0\), which simplifies to \(x > 5\).
This principle ensures that logarithmic functions don't produce undefined or complex numbers, which aren't usable in solving typical real-number equations. Any found solution must respect these domain constraints, rejecting any \(x\) values that don't fit within it.
Quadratic Equations
Quadratic equations often appear unexpectedly when solving logarithmic equations, as they did in our exercise.
After using log properties and simplification, we arrived at an equation: \(x + 2 = 8(x-5)\).
Rearranging these terms led to the quadratic equation \(7x - 42 = 0\).
  • Solve by Simplification: Divide by the coefficient of \(x\), if possible, to further ease solution finding. In our case, dividing by 7, we got:\ \(x = 6\).
Turning complex log expressions into familiar quadratic equations allows for the use of regular algebraic techniques to find solutions. These equations typically follow the standard form \(ax^2 + bx + c = 0\), where efficient methods like factoring, completing the square, or the quadratic formula can then be used.
Solving Equations
Solving equations is the heart of algebra, and with logarithmic equations, this process involves understanding and applying the appropriate mathematical tools and rules. In our exercise, once the logarithmic form was handled, the key steps involved:
  • Apply Logarithmic Rules: Simplify the equation using log properties first.
  • Convert to a Solvable Form: Use the equal logs rule, which indicates \(a^c = b\) if \(\log_a(b) = c\).
  • Transform Into Familiar Equations: Look for forms like quadratic equations that are easier to solve using known methods.
  • Respect Domain Restrictions: Ensure the solutions fit within the domain constraints of the original logarithmic function.
Each step simplifies the complex problem into more straightforward, manageable forms, allowing for solutions that satisfy all mathematical requirements.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After 100 years, a population whose growth rate is \(3 \%\) will have three times as many people as a population whose growth rate is \(1 \%\)

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ Use your graphing utility's power regression option to obtain a model of the form \(y=a x^{b}\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

Use a calculator with \(a\left[y^{x}\right]\) key or \(a \square\) key to solve. India is currently one of the world's fastest-growing countries. By \(2040,\) the population of India will be larger than the population of China; by \(2050,\) nearly one-third of the world's population will live in these two countries alone. The exponential function \(f(x)=574(1.026)^{x}\) models the population of India, \(f(x),\) in millions, \(x\) years after 1974 a. Substitute 0 for \(x\) and, without using a calculator, find India's population in 1974 b. Substitute 27 for \(x\) and use your calculator to find India's population, to the nearest million, in the year 2001 as modeled by this function. c. Find India's population, to the nearest million, in the year 2028 as predicted by this function. d. Find India's population, to the nearest million, in the year 2055 as predicted by this function. e. What appears to be happening to India's population every 27 years?

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ Use your graphing utility's linear regression option to obtain a model of the form \(y=a x+b\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

Graph \(f\) and \(g\) in the same rectangular coordinate system. Then find the point of intersection of the two graphs. Graph \(y=2^{x}\) and \(x=2^{y}\) in the same rectangular coordinate system.
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