Math Problem Statement

The graph in the figure shows the linear relation between log_4 x and log_8 y. Which of the following must be true? Options: A. y^3 = 12x^4, B. y = 12x^2, C. x^2y = 4096, D. y = 4096x^2

Solution

We are solving this problem where the graph describes a linear relationship between log4x\log_4 x and log8y\log_8 y. Let's analyze it step by step:


1. Observations from the graph:

  • The line equation in the (log4x,log8y)(\log_4 x, \log_8 y) plane is a straight line. It passes through:
    • (3,0)(-3, 0) (i.e., when log4x=3\log_4 x = -3, log8y=0\log_8 y = 0),
    • (0,4)(0, 4) (i.e., when log4x=0\log_4 x = 0, log8y=4\log_8 y = 4).

This suggests the line has a slope and intercept that we can use to find the relationship.


2. Finding the slope and equation of the line:

The slope of the line is: slope=ΔyΔx=400(3)=43.\text{slope} = \frac{\Delta y}{\Delta x} = \frac{4 - 0}{0 - (-3)} = \frac{4}{3}.

The line equation in terms of log4x\log_4 x and log8y\log_8 y is: log8y=43log4x+intercept.\log_8 y = \frac{4}{3} \cdot \log_4 x + \text{intercept}.

Using the point (3,0)(-3, 0) to find the intercept: 0=43(3)+intercept    intercept=4.0 = \frac{4}{3}(-3) + \text{intercept} \implies \text{intercept} = 4.

Thus, the equation of the line is: log8y=43log4x+4.\log_8 y = \frac{4}{3} \cdot \log_4 x + 4.


3. Converting the equation into exponential form:

To simplify the relationship, rewrite the logarithms in terms of exponents.

Step 1: Rewrite log4x\log_4 x and log8y\log_8 y in base 2:

log4x=log2xlog24=log2x2,log8y=log2ylog28=log2y3.\log_4 x = \frac{\log_2 x}{\log_2 4} = \frac{\log_2 x}{2}, \quad \log_8 y = \frac{\log_2 y}{\log_2 8} = \frac{\log_2 y}{3}.

Substituting these into the line equation: log2y3=43log2x2+4.\frac{\log_2 y}{3} = \frac{4}{3} \cdot \frac{\log_2 x}{2} + 4.

Multiply through by 3 to eliminate the denominator: log2y=2log2x+12.\log_2 y = 2 \cdot \log_2 x + 12.

Step 2: Combine the terms:

log2y=log2(x2)+log2(212).\log_2 y = \log_2 (x^2) + \log_2 (2^{12}).

Using the properties of logarithms: log2y=log2(x2212).\log_2 y = \log_2 (x^2 \cdot 2^{12}).

Thus: y=x24096.y = x^2 \cdot 4096.


4. Final Answer:

The correct option is D. y=4096x2y = 4096x^2.


Would you like further clarification on any of the steps?

Here are five related questions for practice:

  1. How do you find the equation of a line given two points?
  2. What are the properties of logarithms used in this problem?
  3. How do you convert logarithms of different bases?
  4. How does the slope affect the relationship between variables?
  5. Can you derive the relationship between xx and yy if the slope was different?

Tip: Always use logarithm properties (logabc=clogab\log_a b^c = c \log_a b) to simplify equations!

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Math Problem Analysis

Mathematical Concepts

Logarithms
Linear Equations
Exponential Relationships
Change of Base Formula

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Logarithmic properties: log_a(b^c) = c * log_a(b)
Change of base formula: log_a(b) = log_c(b) / log_c(a)

Theorems

Logarithm and exponentiation relationship: log_a(b) = x ↔ a^x = b

Suitable Grade Level

Grades 10-12