4.5 Exponential and Logarithmic Equations
Ideas in this lesson
- Solve same-base and isolated exponential equations
- Normalize exponentials and use positive substitutions
- Factor polynomial-times-exponential equations
- Solve logarithmic equations with original-domain checks
- Find all numerical roots with graphs and completeness reasoning
- Solve nested and disguised logarithmic equations
- Solve exponential and logarithmic inequalities
- Invert functions and analyze exceptional parameters
- Recover compound-interest time and rate
- Invert contextual exponential models with units and attainable targets
Solve same-base and isolated exponential equations · Worked example
Solve \(3e^{-2x}=9\). Give every real solution exactly and to six decimal places when needed, and check the original equation. Written work is required; selecting an answer alone is not a complete response. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: \(- \dfrac{\ln{\left(3 \right)}}{2}\)
Divide by the coefficient: \(e^{-2x}=\dfrac{9}{3}\).
Taking natural logs gives \(-2x=\ln(\dfrac{9}{3})\).
Divide by negative two: \(x=- \dfrac{\ln{\left(3 \right)}}{2}\).
Substitute the exact value: the exponential becomes the positive ratio and the original coefficient restores \(9\).
The solution is \(x=- \dfrac{\ln{\left(3 \right)}}{2}\approx-0.549306\). These decimals are rounded to six places; the preceding logarithmic values are exact.
Written-work review:
Taking natural logs gives \(-2x=\ln(\dfrac{9}{3})\).
Divide by negative two: \(x=- \dfrac{\ln{\left(3 \right)}}{2}\).
Substitute the exact value: the exponential becomes the positive ratio and the original coefficient restores \(9\).
The solution is \(x=- \dfrac{\ln{\left(3 \right)}}{2}\approx-0.549306\). These decimals are rounded to six places; the preceding logarithmic values are exact.
Written-work review:
- Accept mathematically equivalent derivations and valid alternative counterexamples. Every requested conclusion, original-domain restriction, justification, precision and unit requirement still applies; the listed solution is one acceptable route.
- Required written evidence: Divide by the coefficient: \(e^{-2x}=\dfrac{9}{3}\).
- Required written evidence: Taking natural logs gives \(-2x=\ln(\dfrac{9}{3})\).
- Required written evidence: Divide by negative two: \(x=- \dfrac{\ln{\left(3 \right)}}{2}\).
- Required written evidence: Substitute the exact value: the exponential becomes the positive ratio and the original coefficient restores \(9\).
- Required written evidence: The solution is \(x=- \dfrac{\ln{\left(3 \right)}}{2}\approx-0.549306\). These decimals are rounded to six places; the preceding logarithmic values are exact.