Problem 71 - Precalculus with Calculus Previews

Question

In Problems \(71-76\), find the points on the graph of the given function that have the indicated \(y\) -coordinate. $$ f(x)=6^{x} ; 51 $$

Step-by-Step Solution

Answer

The point on the graph with the given \(y\)-coordinate is approximately \((2.194, 51)\).

Step 1: Understand the Problem

We need to find the value of \(x\) such that the function \(f(x) = 6^x\) equals 51. Essentially, we're solving for \(x\) in the equation \(6^x = 51\).

Step 2: Take the Logarithm of Both Sides

To isolate \(x\), take the logarithm of both sides of the equation \(6^x = 51\). This gives us \(\log(6^x) = \log(51)\).

Step 3: Use Logarithmic Identity

Apply the logarithmic identity \(\log(a^b) = b \log(a)\): \(x \log(6) = \log(51)\).

Step 4: Solve for \(x\)

Rearrange the equation to solve for \(x\): \(x = \frac{\log(51)}{\log(6)}\).

Step 5: Calculate the Value of \(x\)

Use a calculator to find \(\log(51)\) and \(\log(6)\) and divide the results to find \(x\). \[ \log(51) \approx 1.70757 \text{ and } \log(6) \approx 0.77815 \] \[ x \approx \frac{1.70757}{0.77815} \approx 2.194 \].

Step 6: Verify the Result

Check the result by substituting \(x = 2.194\) back into the original function to verify that \(f(x) = 51\) or as close as permissible with rounding. \(f(2.194) = 6^{2.194} \approx 51\).

Key Concepts

Solving Exponential Equations

Exponential equations are equations where the variable is located in an exponent. These equations often look like this: \( a^x = b \). To solve such equations, one common approach is to use logarithms. The goal is to find the value of \( x \) that satisfies the equation.

Here is how the process works:

• First, observe the equation and identify the base of the exponential function, which is "6" in our example \( 6^x = 51 \).
• The next step involves isolating the exponential term, if it's not already isolated.
• At the heart of solving these equations is making use of logarithms to "bring down" the exponent, making it easier to solve for \( x \).

By applying these steps, you can solve any exponential equation, as long as the variable status and the function are clearly defined.

Use of Logarithms

Logarithms are a powerful mathematical tool used to solve exponential equations. They help "transform" these equations into a form that can be more easily handled, particularly when searching for an unknown exponent.

Logarithms work by reversing the operation of exponentiation:

• You start by taking the logarithm of both sides of the equation \( 6^x = 51 \), resulting in \( \log(6^x) = \log(51) \).
• Using the property \( \log(a^b) = b \cdot \log(a) \), the exponent can be brought down in front of the logarithm: \( x \cdot \log(6) = \log(51) \).
• This property allows us to isolate \( x \) because it is no longer trapped in an exponent. Rearrange to give \( x = \frac{\log(51)}{\log(6)} \).

This formula can be used to calculate \( x \) using a calculator which performs logarithmic calculations, leading to a numeric value.

Graph of a Function

The graph of an exponential function like \( f(x) = 6^x \) produces a distinctive curve that rises steeply as \( x \) increases. This results from the rapid increase in the function's values due to the exponentiation.

Key characteristics of this graph include:

• It passes through the point (0,1) because any base raised to the power of zero equals one \( 6^0 = 1 \).
• As \( x \) becomes positive, \( f(x) \) increases dramatically, demonstrating exponential growth.
• Conversely, as \( x \) moves in the negative direction, \( f(x) \) approaches 0 but never touches the x-axis.

Understanding the shape and behavior of this curve is useful because it visually represents how quickly exponential functions can grow, and why logarithms help us find points like \( x = 2.194 \) where \( f(x) = 51 \). By plotting this point on the graph, you can validate the solution visually, seeing \( y = 51 \) on the function's upward trend.