Problem 114 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{10} 4 x-\log _{10}(12+\sqrt{x})=2$$

Step-by-Step Solution

Answer

This can't be calculated without numerical methods, as it leads to a non-algebraic equation. We suggest using some computational software to find the roots, like Wolfram Alpha or a graphing utility to cross-check with the graph.

Step 1: Simplify logarithmic equation

Make use of the property of logarithms that the difference of two logs can be written as a fraction, hence, the equation becomes \( \log_{10} \frac{4x}{12 + \sqrt{x}} = 2 \).

Step 2: Convert logarithmic equation into exponential form

Recall that if \( \log_{b} a = c \), then this is the same as \( b^c = a \). Apply this rule in the equation, thus get: \( \frac{4x}{12 + \sqrt{x}} = 10^2 \), which simplifies to \( \frac{4x}{12 + \sqrt{x}} = 100 \).

Step 3: Clear the fraction

Get rid of the fraction by multiplying every term by \( 12 + \sqrt{x} \), so the equation becomes \( 4x = 100 \times (12 + \sqrt{x}) \).

Step 4: Expand and rearrange

Expand the right side of the equation: \( 4x = 1200 + 100 \times \sqrt{x} \). Rearrange terms to form a quadratic equation: \( 100 \times \sqrt{x} - 4x + 1200 = 0 \).

Step 5: Solve the quadratic equation

To solve for \( \sqrt{x} \), use quadratic formula: \( \sqrt{x} = \frac{-b ± \sqrt{b^2 - 4ac}}{2a} \).

Step 6: Square the solution to get x

Since \( x = (\sqrt{x})^2 \), square both sides of the equation to find the value of x.

Step 7: Verify your answers

Verify the solution by substituting the value of x in the original equation, making sure both sides are equal

Key Concepts

Logarithmic Properties

Logarithmic equations often involve understanding key properties of logarithms which are essential in simplifying complex logarithmic expressions. One fundamental property is that the logarithm of a quotient is equal to the difference of the logarithms, formally written as \( \log_b \left(\frac{a}{c}\right) = \log_b(a) - \log_b(c) \) for any positive real numbers a, c and base b.

Applying this property makes it possible to transform a logarithmic equation with subtraction into one logarithmic expression, which can then be more easily converted into exponential form. Being familiar with these properties is crucial for anyone looking to master solving these types of equations, as seen in the initial simplification step of the given exercise.

Converting Logarithmic to Exponential Form

Transitioning between logarithmic and exponential forms is a powerful technique in solving logarithmic equations. This conversion uses the basic definition of logarithms: if \( \log_b(a) = c \), this translates to the exponential form \( b^c = a \).

Implementing this rule immediately shifts the perspective of the problem, allowing for algebraic manipulations that are more straightforward. In the provided example, once the logarithmic equation is expressed as a single log term, it can be converted into an exponential equation with a base of 10, turning the abstract idea of a logarithm into a more concrete arithmetic equation that we can solve using familiar algebraic methods.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically written in the form \( ax^2 + bx + c = 0 \). The most well-known methods for solving these are factoring, using the quadratic formula, completing the square, or graphically via plotting the parabola.

In our case, after expanding and rearranging, we obtain a quadratic equation in terms of \( \sqrt{x} \), and we use the quadratic formula to solve for \( \sqrt{x} \). Remember, the quadratic formula \( \sqrt{x} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is derived from the general form of a quadratic equation and provides the roots directly. Subsequently, because we're finding \( \sqrt{x} \), we'll need to square the solution to solve for x itself. Understanding how to form and solve quadratic equations is essential when tackling complex logarithmic equations that lead to quadratic forms after logarithmic properties and exponential conversions have been applied.