Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) complete the table to find an interval containing the solution of the equation, (b) use a graphing utility to graph both sides of the equation to estimate the solution, and (c) solve the equation algebraically. Round your results to three decimal places. $$e^{3 x}=12$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & 0.6 & 0.7 & 0.8 & 0.9 & 1.0 \\\\\hline e^{3 x} & & & & & \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The solution to the equation \( e^{3x}=12 \) is approximately 0.891

Step 1: Fill in the table

Put in the values of \( x \) into the exponent equation \( e^{3 x} \) starting from 0.6 to 1.0. Thus the table should look like this: \\ \[\begin{array}{|l|l|l|l|l|l|} \\ \hline x & 0.6 & 0.7 & 0.8 & 0.9 & 1.0 \\ \hline e^{3 x} & e^{1.8} & e^{2.1} & e^{2.4} & e^{2.7} & e^{3.0} \\ \hline \end{array}\]

Step 2: Graph and estimate the solution

Graph the function \( e^{3x} \) and a horizontal line \( y=12 \) using a graphing utility. The point of intersection between these two graphs will provide an approximate solution. Observing the graph and the table, we can say that the solution of this equation lies between 0.8 and 0.9

Step 3: Solve algebraically

Start by taking the natural logarithm \( ln \) on both sides of the equation: \\\[ ln(e^{3x}) = ln(12) \] \\Using the logarithm of a power rule \( ln(a^b)=b*ln(a) \), get: \\\[ 3x*ln(e) = ln(12) \] \\Since \( ln(e)=1 \), this simplifies to: \\\[ 3x = ln(12) \] \\Solving it for \( x \), we get: \\\[ x = \frac{ln(12)}{3}\approx 0.891\]

Key Concepts

Graphing Exponential Functions

Graphing exponential functions is a fundamental skill in algebra that allows students to understand the behavior of exponential growth or decay visually. For the equation e^{3x} = 12, graphing involves plotting the exponential function y = e^{3x} on a coordinate plane.

Start by choosing a range of x values. In this exercise, the range is from 0.6 to 1.0. For each x value, calculate e^{3x} to get the corresponding y value. Then plot these (x, y) points on the plane. The curve of the function will rise steeply, as this is the nature of exponential growth. Additionally, you'll graph the line y = 12, which is a horizontal line across the plane. Where this line intersects the exponential curve is the approximate visual solution to the equation.

Natural Logarithm Applications

The natural logarithm, denoted as ln(x), has countless applications, particularly in solving equations where the variable is an exponent. The natural logarithm is the inverse operation of raising e to a power, which means that ln(e^x) = x.

When confronted with the equation e^{3x} = 12, applying the natural logarithm to both sides allows us to 'bring down' the exponent, making the equation solvable for x. This is thanks to the property that ln(a^b) = b * ln(a). In real-world applications, this property is valuable in fields such as biology to determine population growth, in finance to calculate compound interest, and in physics to deal with exponential decay of radioactive substances.

Algebraic Problem-Solving

Algebraic problem-solving involves manipulating equations and functions to find unknown values. In our exercise, algebraic steps are used to solve an exponential equation, which is initially in the form not easily solvable for the variable x.

First, we recognize the equation is exponential due to the variable residing in the exponent. To isolate x, we utilize properties of logarithms. After taking the natural logarithm on both sides, we use basic algebra to solve for x, dividing both sides by the coefficient of x. Properly understanding algebraic manipulations is crucial, as it allows us to not just rely on graphical approximations but also find precise values that solve our equation.

Exponential and Logarithmic Equations

Solving exponential and logarithmic equations is a balance between understanding the properties of exponents and logarithms and applying algebraic techniques. Exponential equations have variables in the exponents, like e^{3x}, while logarithmic equations involve logarithms with variables inside them.

The two are related by the fact that logarithms are the inverses of exponentials. This relationship is key in solving exponential equations, as taking the logarithm of both sides often simplifies the problem to a form where we can solve for the variable. Remember that logarithms turn multiplication into addition, powers into products, and division into subtraction, which neatly breaks down complex exponential expressions into more manageable algebraic terms.