As a data technician analyzing exponential growth models for your company's logistics software, you are reviewing the algebraic rules used by the system to expand and condense large calculations. Match each property of logarithms to its corresponding mathematical equation.

An inventory analyst is reviewing the algebraic rules used by a logistics program to simplify cost-projection formulas. To expand the logarithm of a product of two positive factors, $M$ and $N$, the analyst must identify the correct Product Property. Which of the following equations accurately represents this property for any valid base $a$?

A software developer is auditing the mathematical functions in a new scientific calculation engine. To optimize the processing of logarithmic data, the developer applies the Inverse Property of logarithms. According to this property, for any valid base $a$ and any positive real number $x$, the expression $a^{\log_a x}$ simplifies to ____.

Analyzing Asset Ratios in Finance

A financial analyst is reviewing a spreadsheet formula used to calculate asset depreciation. The formula applies the Quotient Property of logarithms to simplify a ratio. According to this property, the expression $\log_a \frac{M}{N}$ is equivalent to the quotient of their logarithms, $\frac{\log_a M}{\log_a N}$.