The question reads:

It is given that $\log_5 y$ is a linear function of $\log_{25} x$. The intercepts on the vertical axis and on the horizontal axis of the graph of the linear function are $2$ and $-\frac{2}{3}$, respectively. Which of the following is a constant?

Solution

We are tasked with finding which mathematical relationship results in a constant. Let's break this down:

Step 1: Represent the linear relationship

Since $\log_5 y$ is a linear function of $\log_{25} x$, we can write: $\log_5 y = m \cdot \log_{25} x + c$ where $m$ is the slope, and $c$ is the y-intercept.

Step 2: Use the intercepts

• The y-intercept is given as $2$, which means: $\text{When } \log_{25} x = 0, \quad \log_5 y = 2.$ Thus, $c = 2$.

• The x-intercept is given as $-\frac{2}{3}$, which means: $\text{When } \log_5 y = 0, \quad \log_{25} x = -\frac{2}{3}.$

Step 3: Solve for the slope $m$

Substitute the x-intercept into the linear equation: $0 = m \cdot \left(-\frac{2}{3}\right) + 2.$ Solve for $m$: $m = \frac{2}{\frac{2}{3}} = 3.$

Thus, the equation becomes: $\log_5 y = 3 \cdot \log_{25} x + 2.$

Step 4: Express $x$ and $y$ in terms of base 5

Using the properties of logarithms:

• $\log_{25} x = \frac{\log_5 x}{\log_5 25} = \frac{\log_5 x}{2}$ (since $25 = 5^2$),
• Substituting this into the equation: $\log_5 y = 3 \cdot \frac{\log_5 x}{2} + 2.$ Simplify: $\log_5 y = \frac{3}{2} \cdot \log_5 x + 2.$

Step 5: Combine logarithms

Using the power rule of logarithms ($a \log_b c = \log_b c^a$): $\log_5 y = \log_5 (x^{\frac{3}{2}}) + 2.$ Rewriting: $\log_5 y = \log_5 (x^{\frac{3}{2}}) + \log_5 (5^2).$ Combine logarithms: $\log_5 y = \log_5 (25 \cdot x^{\frac{3}{2}}).$

Thus: $y = 25 \cdot x^{\frac{3}{2}}.$

Step 6: Find the constant

From $y = 25 \cdot x^{\frac{3}{2}}$, the ratio $\frac{y}{x^{\frac{3}{2}}}$ is constant: $\frac{y}{x^{\frac{3}{2}}} = 25.$

Answer:

The constant is $\frac{y}{x^{\frac{3}{2}}}$.

Let me know if you'd like further clarifications or details! Below are 5 related questions:

1. How do you convert logarithms with different bases?
2. What is the significance of the slope in a linear logarithmic relationship?
3. How do intercepts help in determining the equation of a line?
4. What properties of logarithms were used in this problem?
5. How do you determine if a ratio is constant for a given equation?

Tip: Always use the properties of logarithms systematically when solving equations involving multiple bases.