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Chapter 3: Problem 69

Solve each logarithmic equation. 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. $$\log _{3}(x+6)+\log _{3}(x+4)=1$$

Short Answer

Expert verified
The solution to the logarithmic equation \(\log _{3}(x+6)+\log _{3}(x+4)=1\) is \(x=-3\).

Step by step solution

01

Combine the Logarithms

Using the properties of logarithms, we combine the two logarithms into one. The property 'log_a(b) + log_a(c) = log_a(bc)', can be used. Thus, \(\log _{3}(x+6)+\log _{3}(x+4)\) can be written as \(\log_3((x+6)(x+4))\). Now the equation becomes: \(\log_3((x+6)(x+4)) = 1\)
02

Remove the Logarithm

We remove the logarithm by raising both sides as powers of the base 3. That gives us, \((x+6)(x+4) = 3^1\). Since any number raised to 1 is the number itself, this simplifies to: \((x+6)(x+4) = 3\).
03

Simplify the Equation

Simplify the equation by expanding the brackets and rearranging to a standard quadratic form: \(x^2 + 10x + 24 = 3\), which simplifies to \(x^2 + 10x + 21 = 0\).
04

Solve the Quadratic Equation

Now, solve the quadratic equation \(x^2 + 10x + 21 = 0\) by factoring. It can be factored as \((x+3)(x+7) = 0\). The solutions are \(x=-3\) or \(x=-7\).
05

Check the Domain

Substituting both \(x=-3\) and \(x=-7\) into the original logarithmic expressions (\(x+6\) and \(x+4\)) should result in positive numbers, as a negative input is not in the domain of a logarithm. Substituting, we find that \(-3 + 6\) and \(-3 + 4\) are positive, but \(-7 + 6\) and \(-7 + 4\) are not. Thus, \(x=-7\) is not a valid solution and is rejected. Only \(x=-3\) is a valid solution.
06

Decimal Approximation

Since the solution is an integer, no decimal approximation is necessary in this case.

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Key Concepts

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

Properties of Logarithms
Understanding the properties of logarithms is crucial when solving logarithmic equations. Logarithms transform multiplicative processes into additive ones, which is why they are often used to simplify complex equations. The key properties include:
  • The Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \), which allows us to combine two logarithms with the same base when they're being multiplied.
  • The Quotient Rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \), helpful for splitting the logarithm of a division into the difference of two logarithms.
  • The Power Rule: \( \log_b(m^k) = k\log_b(m) \), which lets us bring the power outside of the logarithm as a multiplier.
Applying these properties effectively can greatly simplify the initial equation and is an essential step in finding solutions to logarithmic equations.
Quadratic Equation Solving
Quadratic equations appear frequently in mathematics and are typically of the form \( ax^2 + bx + c = 0 \). There are various methods to solve these equations, including:
  • Factoring: This involves breaking down the quadratic into a product of two binomials. It's efficient when such factorization is readily apparent.
  • Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is a surefire way to find solutions when factoring is not possible.
  • Completing the square: A method that modifies the equation into a perfect square trinomial, making it easier to solve.
Knowing which method to use and executing it correctly are vital for the accurate solution of quadratic equations. Remember, if the discriminant (\( b^2 - 4ac \) in the quadratic formula) is negative, the equation has no real solutions.
Logarithm Domain
The domain of a logarithm is a set of all permissible input values, which stems from the definition of the logarithm itself. Since a logarithm is the inverse function of an exponential, the base, typically denoted as \( b \), must be positive and not equal to 1. Moreover, the argument (the value inside the logarithm, say \( x \)) must also be positive. This is because you cannot take the logarithm of zero or a negative number within the realm of real numbers.
For the equation \( \log_b(x) \), the domain is all \( x > 0 \). It is important to check for any potential solutions against the original equation's domain, as some 'solutions' might be mathematically valid but not in line with the definition of logarithms. Invalid solutions fall outside the domain and must be rejected to ensure the correctness of the final answer.

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Most popular questions from this chapter

Find the inverse of \(f(x)=x^{2}+4, x \geq 0\)

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; \quad 1-\frac{1}{2}+\frac{1}{3} ; \quad 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5} ; \ldots .\) Describe what you observe.

Logarithmic models are well suited to phenomena in which growth is initially rapid but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function.

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

The \(p H\) scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to \(14 .\) A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a pH greater than 7. The lower the \(p H\) below 7 , the more acidic is the solution. Each whole-number decrease in \(p H\) represents a tenfold increase in acidity. (GRAPH CAN'T COPY). The \(p H\) of a solution is given by $$\mathrm{pH}=-\log x$$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve. Express answers as powers of \(10 .\) a. The figure indicates that lemon juice has a pH of 2.3. What is the hydrogen ion concentration? b. Stomach acid has a pH that ranges from 1 to 3. What is the hydrogen ion concentration of the most acidic stomach? c. How many times greater is the hydrogen ion concentration of the acidic stomach in part (b) than the lemon juice in part (a)?
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