Question

To obtain the graph of \(y=\log _{2} 8 x,\) you can either stretch or translate the graph of \(y=\log _{2} x\). a) Describe the stretch you need to apply to the graph of \(y=\log _{z} x\) to result in the graph of \(y=\log _{2} 8 x\). b) Describe the translation you need to apply to the graph of \(y=\log _{2} x\) to result in the graph of \(y=\log _{2} 8 x\).

Short answer

No stretch is needed; Translate the graph of \( y = \log_{2}x \) vertically up by 3 units.

Step-by-step solution

Understanding the basic function

Firstly, understand the basic function. The function given is: \[ y = \log_{2}x \]. This represents a logarithmic function with base 2.

Modify the argument for the stretch transformation

To find out how to transform \( y = \log_{2}x \) into \( y = \log_{2}8x \), express 8 in terms of base 2. \[8 = 2^3\], so, \(8x\) becomes \(2^3 \cdot x\).

Apply properties of logarithms

Using the logarithmic property \( \log_{a}(bc) = \log_{a}b + \log_{a}c \), rewrite the function: \[ y = \log_{2}(8x) = \log_{2}(2^3 \cdot x) = \log_{2}2^3 + \log_{2}x = 3 + \log_{2}x \].

Describe the stretch transformation

Understanding from the previous step, the additional term does not affect the vertical stretching, but rather it shifts the graph upward. Therefore, no vertical stretch is necessary.

Describe the translation change

From step 3, it’s clear that the function is translated vertically. The term \( y = 3 + \log_{2}x \) suggests a vertical translation upwards by 3 units on the y-axis.

Key concepts

logarithmic functions

Logarithmic functions are a type of function that is the inverse of exponential functions. If you have an exponential function like \( y = 2^x \), its logarithmic counterpart is \( y = \log_{2}(x)\). Simply put, the logarithm tells you the power to which you must raise the base (2 in this case) to get a given number. For example, \( \log_{2}(8) = 3 \) because \( 2^3 = 8\). It's crucial to know the common bases used, such as base 10 (common logarithms) and base \( e \) (natural logarithms), but in our case, we always use base 2.

graph transformations

Graph transformations are actions that move or change the shape of your graph. With logarithmic functions, several types of transformations can occur:

• Vertical Translations: Shifting the graph up or down.
• Horizontal Translations: Moving the graph left or right.
• Scaling: Stretching or compressing the graph vertically.
• Reflections: Flipping the graph over an axis.

To get the graph of \( y = \log_{2} (8x)\) from \( y = \log_{2} (x)\), we can apply these transformations. In this specific case, we use vertical translation since our function transforms to \( y = 3 + \log_{2}(x)\).

vertical translation

Vertical translation shifts the graph up or down the y-axis. To visualize, imagine the graph going up like an elevator. From the solution, we see \( y = \log_{2}(8x) \) translates to \( y = 3 + \log_{2}(x).\) Here's why:
First, recognize that \( 8 = 2^3 \), so \( 8x = 2^3 \cdot x\). Using the properties of logarithms, \( y = \log_{2}(8x) \) becomes \( y = \log_{2}(2^3 \cdot x) = \log_{2}(2^3) + \log_{2}(x) = 3 + \log_{2}(x).\)
This process from \( y = \log_{2}(x)\) to \( y = 3 + \log_{2} (x)\) represents a shift of the whole graph three units up.

logarithmic properties

Logarithmic properties help us simplify and understand the transformations better. Some key properties include:

• \textbf{Product Property:} \( \log_{a} (bc) = \log_{a}(b) + \log_{a}(c).\)
• Quotient Property: \( \log_{a} (b/c) = \log_{a} (b) - \log_{a}(c).\)
• Power Property: \( \log_{a} (b^c) = c \cdot \log_{a}(b).\)

In our case, we used the Product Property. By applying it to \( y = \log_{2} (8x) \), we rewrote it as \( y = \log_{2}(2^3 \cdot x) = \log_{2}(2^3) + \log_{2}(x).\)
Then, knowing \( \log_{2} (2^3) = 3 \), we simplified it to \( y = 3 + \log_{2}(x).\). These properties make it easier to perform and understand graph transformations.