Problem 83 - Algebra and Trigonometry

Question

Estimating a Solution Without actually solving the equation, find two whole numbers between which the solution of \(9^{x}=20\) must lie. Do the same for \(9^{x}=100\) . Explain how you reached your conclusions.

Step-by-Step Solution

Answer

The solutions for \(9^x = 20\) lie between 1 and 2, and for \(9^x = 100\) between 2 and 3.

Step 1: Understanding the Equation

We need to estimate the solution for the equations without solving them exactly. The equations given are exponential: \(9^x = 20\) and \(9^x = 100\). We are looking for two whole numbers that the exponent \(x\) will be between.

Step 2: Considering the Powers of 9 for the First Equation

Start by considering whole number values of \(x\) to see where \(20\) would fit. Calculate small powers of \(9\): - \(9^1 = 9\)- \(9^2 = 81\).Since \(9^1 = 9 < 20 < 81 = 9^2\), the solution to \(9^x = 20\) is between \(1\) and \(2\).

Step 3: Verifying the First Estimation

To double-check, consider that \(9^1 \approx 9\) and \(9^{1.5} \approx 30\). Since \(9^1 < 20 < 9^2\), it's confirmed that \(x\) is between \(1\) and \(2\).

Step 4: Considering the Powers of 9 for the Second Equation

For the equation \(9^x = 100\), use similar reasoning:- \(9^2 = 81\)- \(9^3 = 729\).Since \(81 < 100 < 729\), the solution for \(9^x = 100\) must be between \(2\) and \(3\).

Step 5: Verifying the Second Estimation

To ensure this is correct, remember that \(9^2 = 81\) and an estimate like \(9^{2.5} \approx 243\) shows that \(9^2 < 100 < 9^3\). Thus, \(x\) is between \(2\) and \(3\).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent, which can make them a bit tricky to solve. Unlike linear equations, where the variable is a simple coefficient, in exponential equations like \(9^x = 20\) and \(9^x = 100\), the variable \(x\) is in the exponent. This means that the value of the expression grows very rapidly as \(x\) increases.

When you're tasked with estimating solutions, it's about figuring out which two whole numbers this exponent falls between to satisfy the equation without solving it exactly.

To approach this, calculate simpler powers of the base number (in this case, 9) until when the result exceeds the number on the other side of the equation. Then, you'll know that the solution for \(x\) lies between the whole numbers that produced results around that number.

This estimation technique helps understand the equation's behavior and is useful when an exact solution isn't needed or is difficult to find.

Inequality Estimation

Inequality estimation involves finding an estimated range for the solution bound by whole numbers. It is particularly handy when dealing with exponential functions that don't yield easily solvable equations.

For example, consider \(9^x = 20\). By calculating \(9^1 = 9\) and \(9^2 = 81\), you can see that 20 falls between these two products. This estimation tells us \(x\) must lie between 1 and 2. Similarly, for \(9^x = 100\), knowing that \(9^2 = 81\) and \(9^3 = 729\), we see the possible solution for \(x\) lies between 2 and 3.

This method of using inequalities for estimation helps to quickly grasp where solutions might lie and does so without diving into complex calculations or using calculators. It relies on recognizing the growth pattern of exponential functions.

Whole Numbers

Whole numbers are the set of numbers without fractions or decimals, including zero and all positive numbers. In the context of estimating exponential equations, using whole numbers makes it straightforward because we're dealing with simple integer bounds to approximate the value of \(x\).

By observing powers of whole numbers, you achieve a basic understanding of where your solution may fall. For example, by testing \(9^1\) and \(9^2\) for passage solutions like \(9^x = 20\), we determine that the true value of \(x\) should be between 1 and 2 without needing more complex calculations.

The reliance on whole numbers for estimation provides a straightforward technique for constraints when working with exponential expressions, especially in educational contexts where the focus is on understanding the general behavior rather than precise solutions.