Question

Graph \(y=\log _{2} x\) and \(y=\log _{\frac{1}{2}} x\) on the same coordinate grid. Describe the ways the graphs are a) alike b) different

Short answer

Both graphs pass through (1,0) and have vertical asymptotes at x=0. However, \(y = \log_{2} x\) is increasing, while \(y = \log_{1/2} x\) is decreasing.

Step-by-step solution

- Understand the Functions

The two functions to be graphed are logarithmic functions with different bases. The first function is \(y = \log_{2} x\), which has a base of 2. The second function is \(y = \log_{ \frac{1}{2} } x\), which has a base of \frac{1}{2}.

- Graph the Function \(y = \log_{2} x\)

To graph \(y = \log_{2} x\), identify key points. For example, when \(x = 1\), \(y = 0\) because \( \log_{2} 1 = 0\). When \(x = 2\), \(y = 1\). When \(x = 4\), \(y = 2\). Plot these points on the coordinate grid and draw a smooth curve through these points.

- Graph the Function \(y = \log_{ \frac{1}{2} } x\)

For \(y = \log_{ \frac{1}{2} } x\), identify key points. When \(x = 1\), \(y = 0\) because \( \log_{ \frac{1}{2} } 1 = 0\). When \(x = 2\), \(y = -1\) because \( \frac{1}{2}^ { -1 } = 2\). When \(x = 4\), \(y = -2\). Plot these points and draw a smooth curve through these points.

- Compare the Graphs

(a) Alike: Both graphs pass through the point (1,0) because \( \log_{ any-base } 1 = 0\). They both have vertical asymptotes at \(x = 0\) because the logarithm is undefined for non-positive values of \(x\). (b) Different: The graph of \(y = \log_{2} x\) is increasing because it has a base greater than 1, causing the function to grow as \(x\) increases. In contrast, the graph of \(y = \log_{ \frac{1}{2} } x\) is decreasing because the base is between 0 and 1, causing the function to decrease as \(x\) increases.

Key concepts

Logarithm Base Change

Changing the base of a logarithm helps to compare logs with different bases and convert them to a common base. This is important for simplifying complex logarithmic expressions. The formula for changing the base of a logarithm is: \( \text{log}_b a = \frac{\text{log}_k a}{\text{log}_k b} \), where 'b' is the original base, 'a' is the argument, and 'k' is the new base.

For example, to change the base of \( \text{log}_2 8 \) to base 10 (commonly used in calculators), we can write it as \( \text{log}_2 8 = \frac{\text{log}_{10} 8}{\text{log}_{10} 2} \), simplifying to \( \frac{0.9031}{0.3010} \approx 3 \).

Understanding this concept is essential for making calculations easier and comparing different logarithmic functions more effectively.

Graphing Logarithmic Functions

Graphing logarithmic functions involves plotting key points and understanding the behavior of the function based on its base.

For the function \( y = \text{log}_2 x \), it has a base greater than 1, which means the graph will be increasing. Key points include (x=1, y=0), (x=2, y=1), (x=4, y=2). These points help in sketching the curve of the function that starts from the positive side of the x-axis and moves upwards.

For the function \( y = \text{log}_{\frac{1}{2}} x \), the base is between 0 and 1, meaning the graph decreases. Key points here are (x=1, y=0), (x=2, y=-1), (x=4, y=-2). This curve starts from the positive side of the x-axis but decreases as it moves to the right.

Both functions have a vertical asymptote at x=0, meaning they cannot pass through this line as logarithms are undefined for non-positive values.

Properties of Logarithms

Logarithms possess several important properties that make them useful for simplifying expressions and solving equations. Here are some key properties:

• **Product Rule**: \( \text{log}_b (xy) = \text{log}_b x + \text{log}_b y \). This shows that the log of a product is the sum of the logs of the factors.
• **Quotient Rule**: \( \text{log}_b \frac{x}{y} = \text{log}_b x - \text{log}_b y \). This indicates that the log of a quotient is the difference of the logs.
• **Power Rule**: \( \text{log}_b (x^k) = k \text{log}_b x \). This means the log of a power is the exponent multiplied by the log of the base.
• **Change of Base Formula**: \( \text{log}_b x = \frac{\text{log}_k x}{\text{log}_k b} \), essential for converting logs with different bases to a common base.
• **Logarithm of 1**: \( \text{log}_b 1 = 0 \) because any number raised to the power of 0 is 1.
• **Inverse Property**: \( b^{\text{log}_b x} = x \), showcasing that the base raised to its logarithm of x returns x.

Each of these properties plays a crucial role in algebra and calculus, providing powerful tools for simplifying complicated logarithmic expressions and solving real-world mathematical problems.