Points(0.5,6) and (4,24) both lie on the graph of the function y=alog(bx) base of 2. Determine the equation of the function

To find the equation of the function \( y = a \log_2(bx) \) that passes through the points \((0.5, 6)\) and \((4, 24)\), we need to determine the values of \(a\) and \(b\).

Given:
- Point 1: \((x_1, y_1) = (0.5, 6)\)
- Point 2: \((x_2, y_2) = (4, 24)\)
- Function: \( y = a \log_2(bx) \)

### Step 1: Substitute the points into the equation
For the point \((0.5, 6)\):
\[
6 = a \log_2(b \times 0.5)
\]

For the point \((4, 24)\):
\[
24 = a \log_2(b \times 4)
\]

### Step 2: Solve the system of equations
We have the system:
\[
6 = a \log_2(0.5b)
\]
\[
24 = a \log_2(4b)
\]

#### Step 2a: Express \(\log_2(0.5b)\) and \(\log_2(4b)\)
We use the logarithmic identity \(\log_b(xy) = \log_b(x) + \log_b(y)\):
\[
6 = a [\log_2(0.5) + \log_2(b)]
\]
\[
24 = a [\log_2(4) + \log_2(b)]
\]

Knowing that \(\log_2(0.5) = -1\) and \(\log_2(4) = 2\):
\[
6 = a (-1 + \log_2(b))
\]
\[
24 = a (2 + \log_2(b))
\]

#### Step 2b: Simplify and solve for \(a\) and \(\log_2(b)\)
Equation 1:
\[
6 = a(\log_2(b) - 1)
\]
\[
\frac{6}{a} = \log_2(b) - 1
\]
\[
\log_2(b) = \frac{6}{a} + 1
\]

Equation 2:
\[
24 = a(2 + \log_2(b))
\]
\[
\frac{24}{a} = 2 + \log_2(b)
\]
\[
\log_2(b) = \frac{24}{a} - 2
\]

#### Step 2c: Set the two expressions for \(\log_2(b)\) equal
\[
\frac{6}{a} + 1 = \frac{24}{a} - 2
\]

### Step 3: Solve for \(a\)
Rearrange and simplify:
\[
\frac{6}{a} - \frac{24}{a} = -2 - 1
\]
\[
-\frac{18}{a} = -3
\]
\[
\frac{18}{a} = 3
\]
\[
a = 6
\]

### Step 4: Solve for \(b\)
Substitute \(a = 6\) back into the equation for \(\log_2(b)\):
\[
\log_2(b) = \frac{6}{6} + 1 = 1 + 1 = 2
\]
\[
b = 2^2 = 4
\]

### Final Equation
The equation of the function is:
\[
\boxed{y = 6 \log_2(4x)}
\]

Would you like any further details or clarifications?

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Here are 5 related questions to explore:
1. How does the base of the logarithm affect the shape of the graph?
2. What is the effect of changing the coefficient \(a\) in the logarithmic function?
3. How would the function change if the base were \(10\) instead of \(2\)?
4. Can the function \(y = a \log_b(x)\) have negative values? If so, under what conditions?
5. What is the domain of the function \(y = 6 \log_2(4x)\)?

**Tip:** Understanding the properties of logarithms, including their behavior under transformations, can greatly help in solving and graphing logarithmic functions.

Learn how to determine the equation of a logarithmic function \( y = a \log_2(bx) \) that passes through points (0.5, 6) and (4, 24). Understand the process of solving logarithmic equations and identifying function parameters. Suitable for advanced high school students.

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