Solving Exponential and Logarithmic Equations

Task goals

• Foundation: Solve exponential equations by expressing both sides with the same base and equating exponents, and evaluate simple logarithmic equations of the form $\log_a x = b$.
• Proficient: Solve exponential equations with non-matching bases by taking logarithms of both sides, and apply log laws (product, quotient, power) to simplify and solve equations with multiple logarithmic terms.
• Excellence: Solve exponential and logarithmic equations arising from growth/decay or other real-world contexts, including equations requiring substitution (e.g. quadratic-type equations in $e^x$) or combining exponential and logarithmic terms on both sides.