Exponential and Logarithmic Equations & Applications – Study Guidance

Q1. Solve the exponential equation: $2^{3x} = 9$ by graphing.

Background

Topic: Exponential Equations & Graphical Solution

This question tests your ability to solve exponential equations using graphical methods. You are asked to find the value of $x$ that makes $2^{3x}$ equal to $9$ by analyzing the intersection of two functions on a graph.

Key Terms and Formulas

• Exponential Function: A function of the form $f(x) = a^{bx}$, where $a$ and $b$ are constants.

• Graphical Solution: Plotting both sides of the equation as separate functions and finding their intersection point.

Step-by-Step Guidance

1. Rewrite the equation in terms of two functions: $f_1(x) = 2^{3x}$ and $f_2(x) = 9$.

2. Graph both functions on the same set of axes. $f_1(x)$ will be an increasing exponential curve, and $f_2(x)$ will be a horizontal line at $y = 9$.

3. Identify the point of intersection between the curve and the line. The $x$-coordinate of this point is the solution to the equation.

4. Use a graphing calculator or graphing utility to zoom in and estimate the $x$-value where $2^{3x} = 9$.

Try solving on your own before revealing the answer!

Final Answer: $x \approx 1.057$

By graphing, the intersection occurs at $x \approx 1.057$, where $2^{3x}$ equals $9$.

This value can be confirmed by substituting back into the original equation or using logarithms to solve algebraically.