Problem 48 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$8 \log \left(4-x^{2}\right)-4=20$$

Step-by-Step Solution

Answer

No real solution. \(x^2 = -996\) implies no real \(x\).

Step 1: Isolate the Logarithmic Expression

Start by isolating the logarithmic expression in the equation. The given equation is \(8 \log(4-x^2) - 4 = 20\). First, add 4 to both sides to get:\[8 \log(4-x^2) = 24.\] Then, divide by 8 to isolate the logarithm:\[\log(4-x^2) = 3.\]

Step 2: Rewrite Using Exponentiation

Now that the logarithm is isolated, rewrite the equation in exponential form. The expression \(\log(4-x^2) = 3\) implies\[10^3 = 4-x^2.\] Simplify the right-hand side to find:\[1000 = 4-x^2.\]

Step 3: Solve for \(x\)

Rearrange the equation to solve for \(x\):\[4 - x^2 = 1000.\] Subtract 4 from both sides:\[-x^2 = 996.\] Then, multiply both sides by -1 to find:\[x^2 = -996.\] Since \(x^2\) cannot be negative, this equation has no real solutions.

Key Concepts

Exponential Form

When dealing with logarithmic equations, converting them into exponential form is a powerful technique. Understanding how to transition between these forms is crucial. If you have a logarithm, such as \( \log_b(a) = c \), it's telling you that \( b^c = a \).
This conversion moves you from a logarithmic equation to one involving exponents.
In our exercise, once we isolated the logarithm, we had \( \log(4-x^2) = 3 \). Applying the concept of exponential form, this is translated to \( 10^3 = 4-x^2 \).
Thus, we can now think of the problem not in terms of logarithms, but rather with exponents.
This transformation simplifies the process of finding solutions.

Isolating Logarithms

Isolating logarithms is often the first step when solving equations involving them. It involves getting the logarithmic part of the equation on its own on one side. This is usually done to simplify the task of applying further mathematical operations.
In the problem at hand, the initial equation \( 8 \log(4-x^2) - 4 = 20 \) required manipulation to clear the way for solving.

• Add 4 to both sides gave us \( 8 \log(4-x^2) = 24 \).
• Dividing both sides by 8 isolated the logarithm: \( \log(4-x^2) = 3 \).

By isolating the logarithmic component, you simplify the equation, setting up the opportunity for effective resolution of the problem.
Think of isolation as decluttering before diving deeper into solving the problem.

Solving Equations

Once you've navigated to a cleaned-up equation—in either logarithmic or exponential form—it's time to solve for the unknown variable. The core objective is to manipulate the equation into a form where your variable stands alone.
From our expression \( 10^3 = 4-x^2 \), the next step was to solve for \( x \).

• Start by recognizing that \( 10^3 = 1000 \), therefore \( 1000 = 4 - x^2 \).
• Rearrange to \( 4 - x^2 = 1000 \), then subtract 4 from 1000.
• This gives \( -x^2 = 996 \). To isolate \( x^2 \), multiply by -1.
• This results in \( x^2 = -996 \), which needs careful consideration due to the negativity.

The goal is solving for \( x \) while obeying the laws of mathematics, especially pertaining to negatives under a square term.

No Real Solutions

In mathematics, some equations have no solutions in the real number system. This usually occurs when dealing with expressions that aren't feasible in real-world scenarios, such as taking a square root of a negative number.
In the example, our final step led to an expression \( x^2 = -996 \). Here lies the issue; \( x^2 \) being negative under real number operations is impossible.
Since \( x^2 \) must be greater than or equal to zero, \( x^2 = -996 \) implies there’s no real number solution.
This becomes a stopping point: recognizing and understanding that some equations may simply not have a possible solution in real numbers. However, solutions might exist in the complex number system.