Problem 104 - 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. $$\log _{12} x^{2}=6$$

Step-by-Step Solution

Answer

The solution is \(x = 1728\). Verification would show that it satisfies the original equation.

Step 1: Convert to Exponential Form

To convert a logarithmic equation into an exponential form, recall that the base of the logarithm becomes the base of the power, the result of the logarithm becomes the exponent, and the argument of the logarithm becomes the equal side of the equation. Thus, the equation \(\log_{12} x^2 = 6\) becomes \(12^6 = x^2\).

Step 2: Calculate the Square Root

After calculating the value of \(12^6\), you get \(x^2 = 2985984\). Then, to solve for \(x\), you want to take the square root of both sides of the equation. This gives two possible solutions, one positive and one negative, since \(x^2\) could have come from both a positive or a negative number. Thus, you get \(x = \pm \sqrt{2985984}\).

Step 3: Round to Three Decimal Places

After taking the square root, the positive solution is \(x = 1728\), and the negative solution is \(x = -1728\). Since logarithms are undefined for negative inputs and \(x\) can't be a negative number in this context, the only valid solution is \(x = 1728\).

Step 4: Verify the Solution

To verify this solution, substitute \(x = 1728\) back into the original equation and check if both sides are equal. Plug this into a graphing utility to visualize if this is indeed the correct solution.

Key Concepts

Exponential Form

Imagine a world where mathematics can transform one form into another, like turning the humble log cabin into a skyscraper. That's what converting a logarithmic equation into an exponential form does! When you have a logarithmic equation like \( \log_{12} x^2 = 6 \), the goal is to visualize it as an exponential equation.

• The base of the logarithm, which is 12 in our case, becomes the base of the exponent.
• The number on the right side of the equation, which is 6, becomes the exponent.
• The expression after the log, \( x^2 \), becomes the expression equal to 12 raised to this power.

So, we translate \( \log_{12} x^2 = 6 \) to the exponential form \( 12^6 = x^2 \). Now, it's like we've entered a magical world where we can work with a more straightforward equation that eventually helps us solve for \( x \). What a powerful transformation!

Square Root

Square roots are like the gentle giants of the math world, helping us to break down squares to their base numbers. Once we've transformed our logarithmic equation into the exponential form \( 12^6 = x^2 \), our next step is to solve for \( x \) by unleashing the power of the square root.

First, calculate \( 12^6 \), which is 2,985,984. We now need to find \( x \) such that \( x^2 = 2,985,984 \).

• Taking the square root of both sides: \( x = \pm \sqrt{2985984} \).
• The square root of 2,985,984 is 1,728.
• However, be cautious! Only the positive value makes sense here because logarithms are not defined for negative numbers. So, our solution is \( x = 1,728 \).

Embracing the square root is like peeling an orange—once the tough outer layers are removed, you get the sweet solution inside!

Graphing Utility

Visualizing a mathematical solution can be like painting a vibrant picture—it's both enlightening and reassuring. A graphing utility serves this exact purpose. After calculating the solution \( x = 1,728 \), we use a graphing tool to check that the solution truly fits within our equation. It’s one thing to solve it on paper but seeing it displayed graphically gives it another dimension.

When you input \( \log_{12} x^2 \) and plot it, you can verify visually where it intersects with the line \( y=6 \). If the graphically obtained intersection point aligns with \( x=1,728 \), we can confidently confirm our answer.

• It helps ensure accuracy and deepens understanding through visualization.
• Errors can be easily identified, and adjustments can be made.

Using graphing utilities is like having a map in uncharted territories—guiding numbers to where they need to be and validating their placement.