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Chapter 7: Problem 113

Solve: \(\quad \log _{3} x+\log _{3}(x+6)=3\)

Short Answer

Expert verified
The solution to the equation is \(x = 3\).

Step by step solution

01

Combine Logs

Using property of logarithms, we can combine the two logarithms: \(\log _{3} x+\log _{3}(x+6) = \log_{3}(x*(x+6))\). This simplifies to \(\log_{3}(x^2+6x) = 3\)
02

Transform to Exponential Form

We can transform the logarithmic equation into an exponential form. This gives: \(3^{3} = x^2 + 6x\), simplifying to \(27 = x^2 + 6x\). Now this equation looks like a standard quadratic equation.
03

Rearrange into Standard Quadratic Form

We rearrange this equation into a standard quadratic form, by subtracting 27 from both sides to set the equation equal to zero. This gives: \(x^2 + 6x - 27 = 0\).
04

Factor the Quadratic Equation

We can solve for x by factoring the quadratic equation: \((x - 3)(x + 9) = 0\).
05

Solve for x

We set each factor equal to zero and solving, we obtain two potential solutions: \(x = 3\) and \(x = -9\). However, the domain of logarithms are \(x > 0\), so the only valid solution is \(x = 3\).

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Key Concepts

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

Properties of Logarithms
Logarithms allow us to simplify expressions and solve equations through a set of properties. One particularly useful property is the product property, which combines two logarithms with the same base:
  • Product Property: \ \( \log_a b + \log_a c = \log_a (b \cdot c) \)
This property helps condense logarithmic expressions into a single logarithm, making it easier to manipulate equations.For example, in the exercise problem, \ \( \log_3 x + \log_3(x+6) = \log_3(x \cdot (x+6)) = \log_3(x^2 + 6x) \).Equally important are some other key properties:
  • Power Property: \ \( \log_a (b^c) = c \cdot \log_a b \)
  • Quotient Property: \ \( \log_a (\frac{b}{c}) = \log_a b - \log_a c \)
Each property is essential for effective manipulation and solution of logarithmic equations. Recognizing these can greatly simplify your work with complex logarithmic forms.
Exponential Form
Transforming logarithmic equations into exponential form is another critical step when solving these types of problems. The fundamental relationship is:
  • If \ \( \log_a b = c \), then \ \( a^c = b \)
This definition allows us to "unjumble" logarithmic equations into a clearer format. In the given exercise, \ \( \log_3(x^2 + 6x) = 3 \) transforms into the exponential form \ \( 3^3 = x^2 + 6x \). Here, the expression becomes much more straightforward to handle.The exponential form provides a direct pathway to terms that we are more familiar working with, especially when approaching solutions in algebraic equations.
Quadratic Equations
Once a logarithmic equation is expressed in exponential form, it often leads us to a quadratic equation. Quadratic equations generally take the form:
  • Standard Form: \ \( ax^2 + bx + c = 0 \)
Solving quadratic equations can be approached in several ways, primarily through factoring, completing the square, or using the quadratic formula. In the exercise, transforming the equation \ \( 3^3 = x^2 + 6x \) led to \ \( x^2 + 6x - 27 = 0 \), which is in the standard quadratic form.Factoring is often the simplest method, if applicable:
  • For \ \( (x - 3)(x + 9) = 0 \), solve each factor separately \ \( x - 3 = 0 \) and \ \( x + 9 = 0 \: \)
    • Giving potential solutions of \ \( x = 3 \) and \ \( x = -9 \)
  • Remember, solutions must be within domain constraints, especially for logarithms like \ \( x > 0 \)
Understanding these techniques and constraints is valuable for effectively addressing various algebraic challenges.

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Most popular questions from this chapter

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. $$y \geq x^{2}-4$$

Make a rough sketch in a rectangular coordinate system of the graphs representing the equations in each system. The system, whose graphs are a line with negative slope and a parabola whose equation has a negative leading coefficient, has one solution.

Bottled water and medical supplies are to be shipped to survivors of an earthquake by plane. The bottled water weighs 20 pounds per container and medical kits weigh 10 pounds per kit. Each plane can carry no more than \(80,000\) pounds. If \(x\) represents the number of bottles of water to be shipped per plane and \(y\) represents the number of medical kits per plane, write an inequality that models each plane's \(80,000\) -pound weight restriction.

Harsh, mandatory minimum sentences for drug offenses account for more than half the population in U.S. federal prisons. The bar graph shows the number of inmates in federal prisons, in thousands, for drug offenses and all other crimes in 1998 and \(2010 .\) (Other crimes include murder, robbery, fraud, burglary, weapons offenses, immigration offenses, racketeering, and perjury.) (GRAPH CAN'T COPY) a. In \(1998,\) there were 60 thousand inmates in federal prisons for drug offenses. For the period shown by the graph, this number increased by approximately 2.8 thousand inmates per year. Write a function that models the number of inmates, \(y,\) in thousands, for drug offenses \(x\) years after 1998. b. In \(1998,\) there were 44 thousand inmates in federal prisons for all crimes other than drug offenses. For the period shown by the graph, this number increased by approximately 3.8 thousand inmates per year. Write a function that models the number of inmates, \(y,\) in thousands, for all crimes other than drug offenses \(x\) years after 1998 . c. Use the models from parts (a) and (b) to determine in which year the number of federal inmates for drug offenses was the same as the number of federal inmates for all other crimes. How many inmates were there for drug offenses and for all other crimes in that year?

Solve each system by the method of your choice. $$\left\\{\begin{aligned} x^{3}+y &=0 \\ 2 x^{2}-y &=0 \end{aligned}\right.$$
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