Logarithmic Equations and Step-by-Step Algebraic Solutions

Problem Formulation and Multiple-Choice Options

• Problem Objective: Solve for the variable $x$.
• Mathematical Equation Displayed: $\log(3 - x) = 2$
• Numerical Header Indicator: $52$
• Isolated Character Indicator: A
• Multiple-Choice Alternatives:
  • Option A: $0$
  • Option B: $-12$
  • Option C: $-22$
  • Option D: $8$
  • Option E: "Why are lett in math?" (colloquial distractor questioning algebraic variable representation)

Mathematical Principles and Step-by-Step Resolution

• Logarithmic Definition and Exponential Form:
  • A general logarithmic equation $\log_b(y) = c$ is defined such that the base $b$ raised to the power $c$ equals $y$: $b^c = y$.
  • The domain of a real logarithmic function requires the argument to be strictly positive: $y > 0$.
• Evaluation with Base $5$ (Yielding Option C):
  • When the base is $5$, the equation is expressed as $\log_5(3 - x) = 2$.
  • Rewriting into equivalent exponential form gives $3 - x = 5^2$.
  • Evaluating the exponent: $5^2 = 25$.
  • Solving the resulting linear equation for $x$:
  • $3 - x = 25$
  • $-x = 25 - 3$
  • $-x = 22$
  • $x = -22$
  • Domain validation:
  • The argument must satisfy $3 - x > 0$, which implies $x < 3$.
  • Substituting $x = -22$ into the argument yields $3 - (-22) = 25 > 0$.
  • Since $\log_5(25) = 2$, $x = -22$ is a valid solution corresponding to Option C.
• Analysis of Other Bases and Options:
  • Base $10$ (Common Logarithm): If evaluated under standard common log base $10$, $3 - x = 10^2 = 100$, yielding $x = -97$, which is not listed among the choices.
  • Base $2$: If evaluated under base $2$, $3 - x = 2^2 = 4$, yielding $x = -1$, which is not listed among the choices.
  • Option A ($x = 0$): Substituting $x = 0$ produces an argument of $3 - 0 = 3$, which would require a base of $b = \sqrt{3}$ to satisfy $b^2 = 3$.
  • Option B ($x = -12$): Substituting $x = -12$ produces an argument of $3 - (-12) = 15$, requiring a base of $b = \sqrt{15}$.
  • Option D ($x = 8$): Substituting $x = 8$ produces an argument of $3 - 8 = -5$. Because the logarithm of a negative number is undefined within the real number system, $x = 8$ cannot produce a real solution.

Media Context and Engagement Data

• Caption Description: "this is how you simplify log rhythms" (phonetic wordplay on logarithms).
• Interface and Engagement Metrics:
  • Engagement value: $30.1\text{K}$
  • Engagement value: $201$
  • Engagement value: $7,464$
  • Supplementary numeric tag: $523$
  • Interface symbol: $+$