Question

Use a graphing utility to solve the equation. Approximate the result to three decimal places. Verify your result algebraically.\(2^{x}-7=0\)

Short answer

The approximation to three decimal places of the solution for the equation \(2^{x}-7=0\) is \(x \approx 2.807\).

Step-by-step solution

Understanding the problem

Given the equation \(2^{x} - 7 = 0\), we need to solve for \(x\). According to the properties of logarithms, in an equation where the variable is an exponent, taking the logarithm on both sides can simplify the problem.

Rewrite using logarithms

Rewrite the equation using the logarithmic notation. Add `7` to both sides of the equation \(2^{x} - 7 = 0\) to isolate \(2^{x}\) on one side. That gives us \(2^{x} = 7\). Next, take the logarithm base 2 on both sides, which gives us \(x = \log_{2}7\). Depending on your calculators' abilities to calculate logarithms of various bases, you may need to use the logarithm base conversion formula \(x = \frac{\log 7}{\log 2}\).

Approximate the result using a graphing utility

Enter the equation \(x = \frac{\log 7}{\log 2}\) into a graphing utility to find the numerical approximation to three decimal places. It yields a decimal approximation of \(x ≈ 2.807\).

Verify the result algebraically

To verify the result, substitute \(x ≈ 2.807\) back into the original equation \(2^{x}-7=0\). This gives us \(2^{2.807} - 7 \approx 0\). Hence, the solution is verified.

Key concepts

Using a Graphing Utility

A graphing utility is an essential tool in solving equations, especially when dealing with complex functions or equations that are difficult to solve analytically. When an equation includes an exponential term such as in the equation \(2^{x} - 7 = 0\), graphing the function can provide a visual representation of where the function crosses the x-axis, which indicates the solution.

To use a graphing utility to solve an exponential equation, you need to plot the function \(y = 2^{x} - 7\) and observe the point where it intersects the x-axis. The x-coordinate of this intersection point is the solution to the equation. If your graphing utility has the capability, you can use it to directly compute an approximation for the resulting value of \(x\). The utility often includes options to zoom in and analyze the graph to achieve an approximation up to the desired number of decimal places.

Logarithmic Notation

Logarithmic notation is vital for solving equations where the variable is an exponent, as is the case with exponential equations. In the equation \(2^{x} - 7 = 0\), logarithms help us to isolate \(x\) and make the equation solvable. A logarithm has two main parts: the base and the argument. When the equation is rewritten in logarithmic form as \(x = \log_{2}7\), we are stating that \(x\) is the power to which the base \(2\) must be raised to obtain the argument \(7\).

The logarithm tells us about the relationship between exponentiation and its inverse function, the logarithm itself, which allows us to reframe exponential growth or decay in a more manageable way. Knowing how to properly read and write logarithmic notation is a fundamental skill in mathematics that enables students to manipulate and solve various types of equations.

Logarithm Base Conversion

The base conversion formula for logarithms is another powerful tool when solving exponential equations. It allows us to convert a logarithm of one base to a logarithm of another base, which is handy when your calculator can only handle natural logarithms (base \(e\)) or common logarithms (base \(10\)). The formula is given by \(\log_{b}a = \frac{\log a}{\log b}\), where \(\log\) could represent either the natural or common logarithm.

In the case of solving \(2^{x} = 7\), you can apply this formula when we need to find \(x\) and no logarithm base \(2\) button is available on the calculator. The base conversion formula allows you to use whatever base is available, commonly base \(10\), and correctly compute logarithms of base \(2\), thus enabling you to solve \(x = \frac{\log 7}{\log 2}\) with relative ease.

Algebraic Verification

After calculating an approximation for \(x\) using a graphing utility or logarithms, it's important to verify the solution algebraically to confirm its accuracy. Algebraic verification involves substituting the calculated value back into the original equation to see if it satisfies the equation. This step ensures that the solution derived graphically or through logarithms is indeed correct.

For our example equation \(2^{x} - 7 = 0\), we'll take the approximate solution \(x ≈ 2.807\) and plug it back into the function: \((2^{2.807} - 7)\). We need to make sure that this expression is approximately equal to zero. If the left-hand side equates to or is very close to zero, it indicates that the calculated solution is valid. This reassurance is not only satisfying but necessary, as it is the mathematical proof of the accuracy of our result.