Problem 57 - Algebra and Trigonometry

Question

Use the Change of Base Formula to show that $$ \log _{3} x=\frac{\ln x}{\ln 3} $$ Then use this fact to draw the graph of the function \(f(x)=\log _{3} x .\)

Step-by-Step Solution

Answer

Use the Change of Base Formula to express \(\log_3 x\) as \(\frac{\ln x}{\ln 3}\), then graph \(f(x)=\log_3 x\) using this form.

Step 1: Understand the Change of Base Formula

The Change of Base Formula for logarithms states that for any positive numbers \(a\), \(b\), and \(x\), \(\log_b a = \frac{\log_c a}{\log_c b}\). This formula allows us to convert logarithms of different bases to a common base, usually 10 (common logarithm) or \(e\) (natural logarithm).

Step 2: Apply the Change of Base Formula

To express \(\log_3 x\) using natural logarithms, apply the Change of Base Formula: \(\log_3 x = \frac{\ln x}{\ln 3}\). Here, \(c = e\), thus using natural logarithms (\(\ln\)).

Step 3: Confirm the Conversion is Correct

Verify that the expression \(\frac{\ln x}{\ln 3}\) correctly represents \(\log_3 x\). The right-hand side evaluates to the same power that transforms 3 into \(x\), satisfying the definition of logarithms as exponents.

Step 4: Graphing the Logarithmic Function

The function \(f(x) = \log_3 x\) can now be graphed using the converted form \(f(x) = \frac{\ln x}{\ln 3}\). This graph is a logarithmic curve that passes through the point (1,0) because \(\log_3 1 = 0\) and approaches negative infinity as \(x\) approaches 0 from the positive side.

Key Concepts

Logarithms

Logarithms are the mathematical operation that helps us find the power or exponent needed to get a particular number from a certain base. Essentially, if you have an equation like \( a^b = c \), the logarithm of \( c \) to the base \( a \) is \( b \). This is written as \( \log_a c = b \).
Understanding logarithms simplifies complex multiplication and division, as it allows us to work with addition and subtraction instead. This is why logarithms are very useful in fields like computer science, engineering, and finance.
To easily switch from one base to another, you can use the Change of Base Formula. This formula helps to express logarithms in terms of a base that is more convenient to work with, often converting to base 10 or \( e \). The formula is given by \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) could be any positive number.

Natural Logarithms

Natural logarithms are a special type of logarithm where the base is the irrational number \( e \), approximately 2.718. This base is very common in mathematics because it appears in a lot of natural growth processes, such as population growth and radioactive decay.
The natural logarithm of a number \( x \) is written as \( \ln x \). All the properties of regular logarithms apply to natural logarithms as well. For instance, for multiplication inside the logarithm, you can use \( \ln (xy) = \ln x + \ln y \). Similarly, for division, \( \ln \left(\frac{x}{y}\right) = \ln x - \ln y \).
In our exercise, we converted \( \log_3 x \) to natural logarithm form \( \frac{\ln x}{\ln 3} \) using the Change of Base Formula. This step makes it easier to graph the function on a calculator or a computer that handles natural logarithms more efficiently.

Graph of Logarithmic Function

To understand how the graph of a logarithmic function looks, consider the function \( f(x) = \log_3 x \). Once we express it using the Change of Base Formula, this becomes \( f(x) = \frac{\ln x}{\ln 3} \). This is especially useful for graphing because many graphing tools easily handle natural logs.
The graph of \( f(x) = \log_3 x \) is a smooth curve that:

• Passes through the point \( (1,0) \) since \( \log_3 1 = 0 \).
• Decreases towards negative infinity as \( x \) approaches zero from the right, meaning it is undefined for \( x \leq 0 \).
• Grows slowly and indefinitely as \( x \) increases, reflecting the fact that logarithms grow much slower than linear or exponential functions.

By understanding these characteristics, you gain insight into the relationship between exponential and logarithmic growth. The curve's shape tells us how quickly values change, which is essential in interpreting data and predicting trends.