Problem 20 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically. $$\begin{aligned}&f(x)=3 \log _{5} x\\\&g(x)=6\end{aligned}$$

Step-by-Step Solution

Answer

The solution to the equation \(f(x) = g(x)\) is \(x = 25\).

Step 1: Graph the Functions

Use a graphing utility to graph both the functions, \(f(x) = 3 \log_5 x\) and \(g(x) = 6\). Plot them on the same viewing window for comparison.

Step 2: Approximate Intersection

Find the intersection point of the two graphs. This intersection point indicates the value of \(x\) at which \(f(x) = g(x)\). Approximate this value graphically. It will be where the two graphs cross each other.

Step 3: Solve Algebraically

Now, for a more accurate solution, we'll solve \(f(x) = g(x)\) algebraically. So, set \(f(x) = g(x)\), which implies \(3 \log_5 x = 6\). From there, solve for \(x\): divide by 3 on both sides to get \(\log_5 x = 2\). Remember that since 5 squared is 25, \(x = 25\).

Key Concepts

Logarithmic Functions

Logarithmic functions are a special type of mathematical function used to solve for an exponent. These functions are the inverse of exponential functions. The general form of a logarithmic function is \(y = \log_b x\), where \(b\) is the base, \(x\) is the value you're taking the log of, and \(y\) is the exponent. For example, if you have \(\log_5 25 = 2\), this means that \(5^2 = 25\).
If you are asked to solve problems involving logarithmic functions, remember these basics:

• The base of a logarithm tells you the power to raise it to in order to get the number inside the log.
• Logarithmic functions can transform exponentially growing patterns into more manageable linear forms.
• It's crucial to understand the relation between log functions and their bases to solve equations efficiently.

In our problem, the logarithmic function is \( f(x) = 3\log_5 x\). Here, your task is to find the \(x\) that makes this logarithmic expression equal to 6 when compared to another function, \(g(x)\).

Graphing Utilities

Graphing utilities are digital tools that help visualize mathematical functions. They are invaluable when trying to understand complex functions, like logarithms, and are perfect for spotting the relationships and intersections in equations systems. These utilities can range from simple graphing calculators to advanced computer software.
When using a graphing utility:

• Input your functions correctly: Ensure that the equation format in the utility matches your mathematical expression.
• Adjust your viewing window: Sometimes, adjusting how much of the graph you can see helps in identifying intersections more clearly.
• Zoom in on intersections: This makes it easier to precisely estimate points where the graphs intersect.

In the given exercise, using a graphing utility allowed you to plot \(f(x) = 3\log_5 x\) and \(g(x) = 6\). By viewing both on the same window, we could approximate their point of intersection.

Algebraic Solutions

Solving equations algebraically means finding an exact answer using algebraic manipulations and properties. This method can complement graphical approximations by offering precise solutions.
Steps for solving algebraically:

• Set up the equation: Begin by writing down the equation where the two functions are equal, such as \(3 \log_5 x = 6\).
• Isolate the logarithmic expression: Simplify this equation, often by eliminating coefficients or constants—here, divide both sides by 3.
• Change to exponential form: After simplifying, convert the remaining expression using its inverse operation—switch \(\log_5 x = 2\) to \(5^2 = x\).
• Calculate: Solve for \(x\) using basic arithmetic.

Therefore, our example concludes with \(x = 25\), replacing the approximated solution with an exact one.

Points of Intersection

Points of intersection occur where two graphs meet, visually representing equality between functions in a common domain. Finding these points is essential in understanding where two expressions evaluate to the same value.
How to find points of intersection:

• Graphically: By graph plotting, spots where graphs intersect are evident; these points often correspond to solutions of the equation set \(f(x) = g(x)\).
• By data: Some graphing utilities offer tracing features that help zero in on exact intersection coordinates.
• Algebraically confirm: You can verify your findings by further algebraic methods for accuracy assurance.

In this task, the intersection point of \(3 \log_5 x\) and \(g(x) = 6\) suggests the \(x\)-value making the expressions equivalent. Graphs indicated this intersected around \(x = 25\), which aligned with the algebraic solution.