Problem 87 - Algebra and Trigonometry

Question

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. $$ \log x+\log (x+3)=\log 10 $$

Step-by-Step Solution

Answer

The solution to the logarithmic equation is \(x=2\).

Step 1: Use the 'logarithmic product rule'

According to the 'logarithmic product rule', when two logarithms with the same base have an addition operation between them, then they can be written as a single log function where the inputs are multiplied together. Therefore, transform the equation using this rule: \[ \log{x} + \log{(x+3)} = \log{x*(x+3)} = \log{10}.\] For a clearer view, you can represent \(x*(x+3)\) as \(x^2 + 3x\). Therefore, your equation now looks like this: \[ \log{x^2 + 3x} = \log{10}.\]

Step 2: Cancel out the log on both sides

When you have the same logarithm with the same base on both sides of the equation, you can discard the log and keep the argument. Therefore, the equation becomes: \[x^2 + 3x = 10.\]

Step 3: Solve the quadratic equation

This is a quadratic equation, rewrite it as: \[x^2 + 3x - 10 = 0.\] In order to find its solutions, employ the quadratic formula \(x = [-b \pm \sqrt{(b^2 - 4ac)}]/2a.\] Considering a = 1, b = 3, and c = -10, the solutions for x would be: \[x = [-3 \pm \sqrt{(9 + 40)}]/2\]

Step 4: Calculate x values

Calculate the x values: \[x = [-3 \pm \sqrt{49}]/2 = [-3 \pm 7]/2.\] Therefore, the solutions are \(x_1 = 2, x_2 = -5\).

Step 5: Reject invalid x value

Since a logarithmic function is only defined for positive values of the input variable, reject any negative x value. $x=2$ is the only acceptable solution for this problem: \(x=2\).

Key Concepts

Logarithmic Product Rule

Understanding the logarithmic product rule is crucial when solving logarithmic equations. Essentially, this rule states that the sum of two logarithms with the same base can be combined into a single logarithm by multiplying their arguments. In other terms, if you have \(\log_a(b) + \log_a(c)\), you can combine them to \(\log_a(bc)\). This property stems from the fundamental nature of logarithms as exponents and their relationship with multiplication in the base's power.

Let's consider the initial part of our equation \(\log x + \log (x+3)\). By applying the logarithmic product rule, these two terms become \(\log (x(x+3))\), which simplifies the problem and sets the stage for further manipulation of the equation. It's a powerful tool that can streamline the solving process by transforming a potentially complicated equation into a simpler form.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, which means they have the highest exponent of two for the variable. The general form of a quadratic equation is \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. These equations can be solved using a variety of methods including factoring, completing the square, graphing, or utilizing the quadratic formula.

In our problem, after using the logarithmic product rule, we arrive at the equation \(x^2 + 3x = 10\), which we then rewrite to \(x^2 + 3x - 10 = 0\) to fit the standard quadratic equation form. At this point, to find the values of \(x\) that satisfy the equation, you need to determine which method to use for solving. The quadratic formula is a reliable method that will always provide the solutions, given that the equation is quadratic.

Quadratic Formula

The quadratic formula is a fail-safe method to find the solutions of a quadratic equation when the other methods might be cumbersome or infeasible. It is represented as \(x = [-b \pm \sqrt{(b^2 - 4ac)}]/2a\). Applied to a quadratic equation in the form of \(ax^2 + bx + c = 0\), it directly yields the solution by substituting the coefficients \(a\), \(b\), and \(c\).

Considering our problem \(x^2 + 3x - 10 = 0\), where \(a=1\), \(b=3\), and \(c=-10\), we plug these into the formula as follows: \(x = [-3 \pm \sqrt{(9 + 40)}]/2\), which simplifies to \(x = [-3 \pm 7]/2\), providing us with two potential solutions, \(x_1 = 2\) and \(x_2 = -5\). However, because logarithms are undefined for non-positive numbers, we reject \(x_2 = -5\) and accept \(x=2\) as the only valid solution. The quadratic formula not only finds the solutions for us but, coupled with our understanding of logarithms, allows us to immediately identify the valid solution.