/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 165 Solve: \(\log x+\log (x+1)=\log ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 165

Solve: \(\log x+\log (x+1)=\log 12\) (Section 3.4, Example 8)

Short Answer

Expert verified
The solution to the given logarithmic equation \( \log x+\log (x+1)=\log 12 \) is \( x = 3 \)

Step by step solution

01

Combine Logs

Combine the logs on the left side of the equation using log property \(\log a + \log b = \log ab\). Thus we have: \(\log x*\log (x+1)=\log 12\). That simplifies to: \(\log (x(x+1))=\log 12\)
02

Use Logarithm Property

Next use the log property \( \log a = \log b \) implies \( a = b \). This leaves us with the equation \(x(x+1)=12\)
03

Simplify the Equation

Simplify the equation to quadratic. Given \(x(x+1)=12\), this simplifies to \(x^2 + x - 12 = 0 \)
04

Solve Quadratic Equation

Finally, solve the quadratic equation \(x^2 + x - 12 = 0 \). Factoring the quadratic equation gives us: \((x-3)(x+4) = 0\). By setting each factor equal to zero and solving, we get two results: \(x=3\) and \(x=-4\)
05

Check answers in original equation

Check both possible answers with the original equation, \(\log x+\log (x+1)=\log 12\). The result for \(x=-4\) is undefined, as there's no logarithm for negative numbers. However, for \(x=3\), both sides of the equation are equal. So, the 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
The properties of logarithms are incredibly useful for solving equations that involve logarithmic functions. A fundamental property used in solving logarithmic equations is the product property, which states that ewline ewline ewline ewline ewline Product Property: \( ewline ewline ewline ewline ewline log_a (mn) = ewline ewline ewline ewline ewline log_a (m) + ewline ewline ewline ewline ewline log_a (n) \).ewline ewline ewline ewline ewline
ewline ewline ewline ewline ewline This allows us to combine or decompose logarithmic expressions. Another important property is that of equivalence where if \( ewline ewline ewline ewline ewline log_a (x) = ewline ewline ewline ewline ewline log_a (y) \), then \( ewline ewline ewline ewline ewline x = ewline ewline ewline ewline ewline y \). This property is critical when the goal is to isolate the variable of interest, as seen in the given exercise.
Quadratic Equations
Quadratic equations are polynomials that have the highest degree of 2 and are of the form \( ewline ewline ewline ewline ewline ax^2 + bx + c = 0 \), where \( ewline ewline ewline ewline ewline a \), \( ewline ewline ewline ewline ewline b \), and \( ewline ewline ewline ewline ewline c \) are constants and \( ewline ewline ewline ewline ewline a \) is not zero. These equations are solvable by various methods, including factoring, completing the square, or using the quadratic formula. Understanding how to manipulate and solve quadratic equations is essential for finding the values of x that satisfy the equation, as was required to solve for x in our exercise.
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions and are defined for positive real numbers. A logarithmic function is typically written as \( ewline ewline ewline ewline ewline y = log_b(x) \), indicating the power to which the base \(ewline ewline ewline ewline ewline b \) must be raised to yield \( ewline ewline ewline ewline ewline x \). One key aspect to remember with logarithmic functions is that they are only valid for positive arguments. This is why, in the step-by-step solution, the answer \( ewline ewline ewline ewline ewline x = -4 \) is excluded as logarithms cannot have negative inputs.
Factoring Quadratics
Factoring quadratics is a method used to solve quadratic equations by expressing them as a product of their linear factors. The standard form of a quadratic equation \( ewline ewline ewline ewline ewline ax^2 + bx + c = 0 \) can often be factored into \( ewline ewline ewline ewline ewline (px + q)(rx + s) = 0 \), where p, q, r, and s are numbers that satisfy \( ewline ewline ewline ewline ewline pr = a \) and \( ewline ewline ewline ewline ewline qs = c \), and \( ewline ewline ewline ewline ewline pq + rs = b \). Once the quadratic is factored, the Zero Product Property can be employed, which states that if the product of two factors is zero, then at least one of the factors must be zero. This property is what enables us to find the solution(s) to the equation, as we did in the fourth step of the exercise solution.

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Most popular questions from this chapter

Use this information to solve Exercises \(135-136 .\) When throwing an object, the distance achieved depends on its initial velocity, \(\bar{v}_{0}\) and the angle above the horizontal at which the object is thrown, \(\theta\). The distance, \(d,\) in feet, that describes the range covered is given by $$ d=\frac{v_{0}^{2}}{16} \sin \theta \cos \theta $$ where \(v_{0}\) is measured in feet per second. You and your friend are throwing a baseball back and forth. If you throw the ball with an initial velocity of \(v_{0}=90\) feet per second, at what angle of elevation, \(\theta,\) to the nearest degree, should you direct your throw so that it can be easily caught by your friend located 170 feet away?

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