Problem 103 - 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. $$\log _{10}\left(z^{2}+19\right)=2$$

Step-by-Step Solution

Answer

The solution to the problem is \( z = ± 9 \).

Step 1: Translate the logarithm into an exponential equation

We know that \( \log _{10}\left(z^{2}+19\right)=2 \) can be expressed as \( 10^2 = z^2 + 19 \) based on the definition of the logarithm.

Step 2: Simplify the exponential equation

Solve the equation, \( 10^2 = z^2 + 19 \), for \( z^2 \). Subtract 19 from both sides of the equation to isolate \( z^2 \). This gives us a quadratic equation: \( z^2 = 10^2 - 19 = 81 \)

Step 3: Solve for z

To solve for \( z \), take the square root of both sides. Remember to consider both the positive and negative square root. \( z = ± \sqrt{81} = ± 9 \)

Step 4: Verify the answer using a graphing utility.

We can confirm the solution graphically by plotting the logarithmic function \( y= \log_{10}(x^2 + 19) \) and the line \( y = 2 \) in the same coordinate plane. The points where the graph of the function intersects the line correspond to the solutions of the equation.

Key Concepts

Logarithmic to Exponential Form

Understanding how to convert a logarithmic equation to its exponential form is vital when solving logarithmic equations. The problem presents us with the logarithmic equation \(\log _{10}(z^{2}+19) = 2\). To translate this into an exponential form, we apply the basic definition of logarithms: if \(\log_b(a) = c\), then \(b^c = a\). Here, our base \(b\) is 10, \(a\) is \(z^{2}+19\), and \(c\) is 2. Thus, the exponential form of the given logarithmic equation is \(10^2 = z^{2} + 19\).
This conversion is crucial as it simplifies the problem to a quadratic equation which is more manageable to solve. Without this step, finding the value of \(z\) would be much more complex.

Solving Quadratic Equations

Once the logarithmic equation is in exponential form, we encounter a quadratic equation: \(z^{2} = 81\). Solving quadratic equations is a fundamental skill in algebra. Every quadratic equation can be written in the standard form \(ax^2 + bx + c = 0\) and solved by a variety of methods including factoring, completing the square, using the quadratic formula, or graphing.
In this problem, we apply the most straightforward method. We want to isolate \(z\) so we take the square root of both sides resulting in \(z = ±\sqrt{81} = ±9\). It's essential to include both the positive and negative roots because both are solutions to the original equation. Including both solutions ensures that we find all possible values for \(z\) that satisfy the given logarithmic equation.

Graphical Verification of Solutions

Graphical verification allows us to see whether our algebraic solutions are correct by using a graphing utility. The steps involve plotting the original logarithmic function alongside the constant that the logarithm equaled—in this case, \(y = 2\). We represent our logarithmic function as \(y = \log_{10}(z^2 + 19)\) and plot it on the same axes as the line \(y = 2\). Where these two graphs intersect, the \(z\)-coordinates of the intersection points are our solutions. For our equation, the solution \(z = ±9\) should appear as points of intersection at \(z = 9\) and \(z = -9\) on the graph. This visual representation not only confirms the accuracy of our algebraic solution but also provides a deeper understanding of the behavior of logarithmic functions and thereby enriches our learning experience.