Problem 68 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 67-68: Approximate the function at the value of \(x\) to four decimal places. $$ h(x)=3 \log _{3}(2 x-1)+7 \log _{2}(x+0.2) ; \quad x=52.6 $$

Step-by-Step Solution

Answer

The approximation of the function at \( x = 52.6 \) is 50.4283.

Step 1: Evaluate the First Logarithmic Term

We start by evaluating the first logarithmic term of the function \[ 3 \log_{3}(2x-1) \] at \( x = 52.6 \). Calculate \( 2x-1 \):\[ 2 \times 52.6 - 1 = 104.2 \]Next, calculate the logarithm:\[ \log_{3}(104.2) \]Using a calculator, we find an approximate value:\[ \log_{3}(104.2) \approx 3.4469 \]Thus, the term evaluates to:\[ 3 \times 3.4469 = 10.3407 \]

Step 2: Evaluate the Second Logarithmic Term

Now, evaluate the second logarithmic term of the function \[ 7 \log_{2}(x+0.2) \] at \( x = 52.6 \).Calculate \( x+0.2 \):\[ 52.6 + 0.2 = 52.8 \]Next, calculate the logarithm:\[ \log_{2}(52.8) \]Using a calculator, we find an approximate value:\[ \log_{2}(52.8) \approx 5.7268 \]Therefore, the term evaluates to:\[ 7 \times 5.7268 = 40.0876 \]

Step 3: Add the Results of Both Terms

Finally, add the results of both logarithmic terms to obtain the approximation of the function.Sum the two terms:\[ 10.3407 + 40.0876 = 50.4283 \]Therefore, the approximation of the function \( h(x) \) at \( x = 52.6 \) is 50.4283.

Key Concepts

Logarithmic Term Evaluation

Logarithmic term evaluation is crucial for solving problems that involve logarithmic functions. Logarithms transform multiplicative relationships into additive ones, simplifying complex calculations. In our exercise, we focus on two separate logarithmic terms within a function: \( 3 \log_{3}(2x-1) \) and \( 7 \log_{2}(x+0.2) \). To evaluate these terms, we start by substituting the given value of \( x = 52.6 \). This results in simpler expressions that can be tackled individually. This step-by-step approach helps break down and understand complex logarithmic functions effectively.

Base-3 Logarithm

The base-3 logarithm \( \log_{3}(y) \) is a specific type of logarithm where the base is 3. It answers the question: "To what power must 3 be raised to produce the number \( y \)?". In our function, the term \( \log_{3}(2x-1) \) was considered. After substituting \( x = 52.6 \), we computed \( 2x - 1 = 104.2 \).

• Using a calculator, we found \( \log_{3}(104.2) \) to approximate 3.4469.
• This approximation helps in proceeding with further calculations and simplifies the expression.

The evaluated result \( 3 \times 3.4469 = 10.3407 \) integrates into our overall function to simplify its understanding.

Base-2 Logarithm

The base-2 logarithm, represented as \( \log_{2}(z) \), is fundamental in many areas of mathematics and computer science due to its relevance with binary systems. It helps determine the power to which 2 is raised to produce \( z \). For our problem, \( \log_{2}(x+0.2) \) is used.

• First, calculate \( x+0.2 \) resulting in 52.8, which transforms the expression into \( \log_{2}(52.8) \).
• Approximating \( \log_{2}(52.8) \) yields about 5.7268.
• This approximation leads to 7 multiplied by 5.7268, equal to 40.0876, which forms a part of the function's solution.

Understanding base-2 logarithms aids in simplifying calculations and grasping the function's behavior at a specific point.

Function Approximation

Function approximation is the process of estimating a function's value at specific points where the function's exact calculation might be complex. In this exercise, the aim was to approximate \( h(x) \) at \( x = 52.6 \). By solving individual logarithmic terms approximately and adding them together, we could find an estimated function value.

• The first term, involving base-3 logarithm, evaluated to approximately 10.3407.
• The second term, involving base-2 logarithm, evaluated to approximately 40.0876.
• Adding these two results provided the approximated function value: 50.4283.

Approximating functions in logs assists in simplifying intricate calculations and provides a practical result useful in real-world applications.