Problem 94 - Algebra and Trigonometry

Question

Solve: \(\log (x+4)-\log (x-2)=\log x\) (Section \(4.4, \text { Example } 8)\)

Step-by-Step Solution

Answer

The solution to the equation \(\log (x+4)-\log (x-2)=\log x\) is \(x=4\).

Step 1: Apply the law of logarithms

Subtract the logarithmic expressions on the left side. This converts the equation to: \(\log(\frac{x+4}{x-2}) = \log x\).

Step 2: Equate the arguments of the logarithms

Since the logarithms are equal to each other, the expressions inside the logs must also be equal. Thus we get the equation: \(\frac{x+4}{x-2} = x\).

Step 3: Solve for x

To solve for \(x\), first get rid of the denominator by multiplying both sides by \(x-2\), yielding \(x+4 = x(x-2)\), which simplifies further to \(x+4 = x^2 - 2x\). Rearranging this gives the quadratic equation \(x^2 - 3x - 4 = 0\). Solving this quadratic equation will give the potential solutions for \(x\).

Step 4: Solve the quadratic equation

This quadratic can be factored to \((x - 4)(x + 1) = 0\). Setting each factor to zero gives the solutions \(x=4\) and \(x=-1\).

Step 5: Check the solutions

Check each solution in the original equation to discard any extraneous solutions. Remember the domain of log function is \(x>0\). Insert \(x=4\) back into the original equation: \(\log(4+4) - \log(4-2) = \log 4\), which simplifies to \(\log 8 - \log 2 = \log 4\). Converting to exponential form confirms that \(4=4\). Therefore \(x=4\) is a valid solution. Substitute \(x=-1\) into the original equation: \(\log(-1+4) - \log(-1-2)\) is undefined because the second term \(\log(-3)\) is not defined in the real numbers. Therefore, \(x=-1\) is an extraneous solution and discarded.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential when solving logarithmic equations. Logarithms, essentially the inverse operation of exponentiation, have unique properties that make it easier to manipulate and solve related equations.

For instance, a key property used in our exercise is the subtraction of logarithms, which equates to the logarithm of the division of their arguments: \[\begin{equation}\log(a) - \log(b) = \log\left(\frac{a}{b}\right).\end{equation}\]Other useful properties include the product property, where the addition of two logarithms equals the logarithm of the product of their arguments, and the power property, which allows us to move the exponent of the argument to the front of the logarithm. These properties are invaluable tools in simplifying and solving logarithmic equations.

Simply put, by knowing and applying these properties, you can transform complex logarithmic expressions into more manageable forms, often allowing you to solve equations without the need for calculators or extensive computations.

Logarithmic Functions

A logarithmic function is an operation that answers the question: To what power must a certain base be raised in order to obtain a given number? The general form of a logarithmic function is \[\begin{equation}\log_b(x) = y,\end{equation}\]which is equivalent to the exponential form \[\begin{equation}b^y = x.\end{equation}\]In the context of our equation-solving exercise, we use the natural logarithm with base 10, which is commonly just expressed as \(\log(x)\).

Domain and Range

The domain of a logarithmic function, which is the set of allowable input values (x), is all positive real numbers, because you cannot take the log of zero or a negative number in real number arithmetic. The range, which is the set of possible outputs (y), is all real numbers, because the log function can produce any real number given a positive input.

The correct understanding of the behavior of logarithmic functions is crucial when it comes to checking the validity of solutions to logarithmic equations. This accounts for why certain solutions can be discarded as extraneous, as they do not fall within the domain of the logarithmic function.

Quadratic Equations

Quadratic equations are second-degree polynomial equations in the form \[\begin{equation}ax^2 + bx + c = 0,\end{equation}\]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). To solve quadratic equations, various methods can be used, among which are factoring, completing the square, using the quadratic formula, or graphing.

Factoring Method

The factoring method involves rewriting the quadratic as a product of two binomial expressions, which was done in the given exercise. Once factored, you can use the zero-product property which states that if the product of two factors is zero, then at least one of the factors must be zero. This leads to finding the potential roots of the equation.

It’s critical to always check the solutions when dealing with logarithmic equations by substituting them back into the original equation to ensure they do not fall outside the domain of the logarithmic function, as seen in our solved example. Applying these methods, along with the properties of logarithms and understanding of logarithmic functions, equips you for solving similar mathematical problems with confidence.