Example 10.38: Solving $2\log_5 x = \log_5 81$

In a technical data verification process, an analyst uses the One-to-One Property of Logarithmic Equations to compare two signal intensities, $x$ and $y$. If the analyst establishes the equation $\log_b(x) = \log_b(y)$ for a valid base $b$, which of the following represents the direct conclusion reached by applying this property?

A structural engineer is using an equation that simplifies to $\log_2(x) = \log_2(y)$ to evaluate stress loads. Applying the One-to-One Property of Logarithmic Equations, the engineer concludes $x = y$. However, as a final procedural step, the engineer must verify these calculated values in the original equation to eliminate any extraneous solutions, because logarithms are mathematically defined only for ____ quantities.

An electronics technician is comparing two signal voltages, $V_1$ and $V_2$, and observes the equation $\log_2(V_1) = \log_4(V_2)$. According to the One-to-One Property of Logarithmic Equations, the technician can conclude that $V_1 = V_2$ because both sides involve a single logarithm.

A business analyst is using logarithmic growth models to determine when two different investments will reach the same value. To solve the resulting equations, the analyst applies the One-to-One Property of Logarithmic Equations. Match each component of the property with its corresponding description or role in the mathematical analysis.

A quality control engineer is monitoring sound intensity levels in a manufacturing plant to ensure they meet safety standards. To find an unknown variable in a noise-level comparison, the engineer must solve an equation using the One-to-One Property of Logarithmic Equations. Arrange the following steps in the correct procedural order to solve the equation.