Subjects algebra

Logarithmic Function 7Ac574

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1. **State the problem:** Simplify or analyze the function given by $$y=\frac{\ln\left(\frac{x}{m}-sa\right)}{r^2}$$ where $x$, $m$, $s$, $a$, and $r$ are variables or constants. 2. **Recall the formula and rules:** The natural logarithm function $\ln(z)$ is defined for $z>0$. The division by $r^2$ means the entire logarithm expression is divided by $r^2$. Important: $r \neq 0$ to avoid division by zero. 3. **Rewrite the function for clarity:** $$y=\frac{\ln\left(\frac{x}{m}-sa\right)}{r^2}$$ 4. **Domain considerations:** - The argument of the logarithm must be positive: $$\frac{x}{m} - sa > 0$$ - Solve for $x$: $$\frac{x}{m} > sa \implies x > m \cdot sa$$ - Also, $r \neq 0$. 5. **No further simplification is possible without values.** **Final answer:** $$y=\frac{\ln\left(\frac{x}{m}-sa\right)}{r^2}$$ with domain $x > m \cdot sa$ and $r \neq 0$.