Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{1 / 3}(x+2)$$.

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\log_{1/3}(x+2)\) is the reflection of the graph of base-e logarithm function over the x-axis, and it is shifted 2 units to the left.

Step 1: Re-write the Function

We first transform the function \(f(x)=\log _{1 / 3}(x+2)\) using the change of base formula to base \(e\) (natural logarithm). According to the change-of-base formula \(\log_{a}{b}=\frac{\ln b}{\ln a}\), our function therefore becomes \(f(x) = \frac{\ln(x+2)}{\ln(1 / 3)}\).

Step 2: Simplify the Function

We can simplify the denominator of the fraction by evaluating the natural logarithm of \(1/3\). Remember that \(\ln(1/3)\) is a negative number, so our function becomes \(f(x) = -\frac{\ln(x+2)}{\ln(3)}\)

Step 3: Graphing the Function

Finally, we graph this function using a graphing utility. Remember that we are working with logarithmic functions, which will generate a curved line. The graph will be a reflection of \(y=\ln(x)\) over the x-axis, and shifted 2 units to the left. This is because we replaced x in the function \(y=\ln(x)\) with \((x+2)\), and we multiplied it with a negative sign.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They help us to solve equations where the variable is an exponent. A logarithmic function can be generally expressed as \( \log_a(x) \), where \( a \) is the base and \( x \) is the argument.

Logarithms simplify the process of multiplication, division, and exponentiation by transforming them into addition, subtraction, and multiplication, respectively. This is particularly useful in scientific calculations. When dealing with bases other than 10 or \( e \), we use the change-of-base formula to convert them. The change-of-base formula allows easy transformations between different logarithmic bases:

• \( \log_a(b) = \frac{\ln b}{\ln a} \)

Understanding logarithms helps us interpret growth patterns, sound intensity, and much more in real-life applications.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is particularly important since its base is \( e \), an irrational constant approximately equal to 2.718. The natural logarithm is widely used in mathematics, physics, and engineering due to the natural base \( e \)'s properties.

It makes the derivative and integral calculations simpler and is essential to the growth processes and decay problems. The natural logarithm is also used in calculating compound interest, measuring entropy, and even in calculating probabilities and statistics. When rewriting a logarithm using the change-of-base formula, the natural logarithm simplifies calculations by relying on this constant base. Utilizing the formula \( \log_a(x) = \frac{\ln x}{\ln a} \), helps to work with any base, converting it into natural logarithmic form.

This aids in using various mathematical functions without changing the core aspects of calculations.

Graphing Utility

Graphing utilities, such as graphing calculators or software like Desmos, help visualize mathematical functions including logarithmic functions. These tools allow for interactive exploration of different functions by presenting their graphical representations.

When graphing the function \( f(x) = \log_{1/3}(x+2) \), we need to apply transformation rules to understand its shape and position. By using a graphing utility:

• You can see how the graph reflects across the x-axis due to the negative in the function \( -\frac{\ln(x+2)}{\ln(3)} \).
• Observe the horizontal shift left, as replacing \( x \) with \( (x+2) \) shifts the function 2 units to the left.

Graphing utilities make it easy to manipulate functions and observe the effect of transformations, offering insightful visual aids for learning complex mathematical concepts.