In a technical math refresher for laboratory analysts, you are reviewing the essential steps to solve logarithmic equations. Match each logarithmic expression or rule with its correct equivalent exponential form or definition.

In a technical math workshop, a student is solving the equation $\log_a 49 = 2$ to determine a calibration base $a$. After converting the equation to its exponential form $a^2 = 49$, the student finds two potential values: $a = 7$ and $a = -7$. Which fundamental property of logarithms must the student recall to correctly identify that $a = -7$ is an extraneous solution?

As part of a technical math workshop for laboratory assistants, you are reviewing the standard procedure for solving equations with unknown bases. Arrange the following steps in the correct order to solve for the base $a$ in the equation $\log_a 49 = 2$.

In a technical math workshop, you are reviewing how to solve logarithmic equations. True or False: The natural logarithmic equation $\ln x = 3$ is equivalent to the exponential equation $x = e^3$.

Solving Natural Logarithmic and Unknown Base Equations

During a manufacturing quality-control check, a technician models a component's degradation rate using the unknown-base logarithmic equation $\log_a 49 = 2$. After converting this to its exponential form $a^2 = 49$, the algebraic solution process yields $a = 7$ and $a = -7$. The technician must reject the solution $a = -7$ because the base $a$ of any logarithmic expression must always be ____.

Environmental Sensor Calibration Log