Problem 41 - Precalculus : Building Concepts and Connections

Question

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log _{3}(x+4)=2$$

Step-by-Step Solution

Answer

The solution to the logarithmic equation \(\log _{3}(x+4)=2\) is \(x=5\).

Step 1: Transforming the Logarithmic Equation

To eliminate the logarithm, convert the logarithmic equation \(\log _{3}(x+4)=2\) into an exponential equation using the identity \(a=\log _{b}(m)\) which is equivalent to \(b^{a}=m\). Thus \(\log _{3}(x+4)=2\) becomes \(3^{2}=x+4\).

Step 2: Solving the Exponential Equation

Now, solve the exponential equation for x. This yields \(3^2 - 4 = x\), which simplifies to \(x = 5\).

Step 3: Checking for Extraneous Solutions

Substitute x = 5 into the original logarithmic equation \(\log _{3}(5+4)=2\). This statement is true, confirming that an x-value of 5 is a valid solution and is therefore not extraneous.

Key Concepts

Exponential Equations

Exponential equations are a type of equation in which a variable appears in the exponent. They often arise when working with logarithmic functions, as transforming a logarithm to an exponential form helps simplify solving the equation. For instance, when given an equation like \(\log_b(m) = a\), you can transform it to exponential form as \(b^a = m\). This transformation is crucial as it allows direct calculations.

• Exponential equations are solved by applying the rules of exponents.
• All terms with the variable exponent should be isolated to facilitate easy calculation.

In our exercise, the logarithmic equation \(\log_3(x+4) = 2\) is converted to an exponential format, \(3^2 = x+4\). This simplification removes the logarithmic component, making the equation easier to solve.

Solving Equations

When solving equations, especially those derived from logarithmic expressions, it's essential to follow systematic steps. After converting a logarithmic equation to exponential form, you can use basic algebraic operations to find the solution for the variable.

• After conversion, isolate the variable by using inverse operations, such as subtraction or division, as applicable.
• Always perform operations symmetrically on both sides of the equation to maintain equality.

In our example, the transformed equation is \(3^2 = x+4\). By subtracting 4 from both sides, \(x = 9 - 4\), which simplifies to \(x = 5\). Skipping any step may lead to incorrect answers, so thoroughly solving each part ensures accuracy.

Extraneous Solutions

Extraneous solutions are potential solutions that arise during the solving process but do not satisfy the original equation. These solutions can occur particularly when dealing with logarithmic and radical equations, as certain transformations may introduce them.

• After obtaining a solution, always substitute it back into the original equation to verify its validity.
• An extraneous solution will not satisfy the original equation, indicating it is not a viable answer.

In the provided problem, the calculated solution of \(x = 5\) is verified by plugging it back into the initial logarithmic equation: \(\log_3(5+4) = 2\). This verification confirms that the statement holds true, ensuring \(x = 5\) is not extraneous. Always remember this checking step to rule out false solutions.