Solving Exponential Equations with Logarithms
Take logarithms of both sides to bring an unknown down from an exponent, and solve equations that are quadratic in eˣ by substitution.
What a learner can do afterwards
- Solve 3ˣ = 20 to three significant figures
- Solve e^(2x) - 5eˣ + 6 = 0 using a substitution
- Find how long a modelled quantity takes to halve
1 · Read
You isolate the exponential part first, then take logs of both sides. The power rule pulls the exponent down front, and that log step is what frees x. Take 3^x = 20: logs give x log 3 = log 20, so x = log 20/log 3, about 2.73. Round only at the very end.
When both sides share a base, skip the logs and match the exponents. Since 125 is 5^3, the equation 5^x = 125 gives x = 3 at once. Use this shortcut whenever the same base appears on both sides.
Treat e^(2x) - 5e^x + 6 = 0 as a quadratic in disguise. Substitute y = e^x to get y^2 - 5y + 6 = 0, which factors to y = 2 or y = 3. Undo the swap with logs: x = ln 2 or x = ln 3, not 2 or 3.
For halving stories, count how many half lives fit and halve that many times. A sample of 80 g with a 4 hour half life gives 3 half lives in 12 hours, so 80 to 40 to 20 to 10 g. A tracer falling from 200 to 25 MBq halves 3 times, and at 6 hours each that takes 18 hours.
Free x with logs, swap e^x for y in quadratic cases, and count half lives in decay stories.
The rewritten readings were written for this dataset.
More to read
Solving exponential equations using logarithms
When the unknown sits in the exponent, logarithms are the tool that brings it down. The standard move is to isolate the exponential part first, then take logs of both sides and use the power rule to pull the exponent out front. If both sides share the same base, you can skip the logs and just equate the exponents. Equations like e to the 2x minus 5 e to the x plus 6 = 0 look scary until you substitute y = e to the x, which turns them into plain quadratics. Always check your answers back in the original equation, since log steps can smuggle in extras.
Choosing an exponential equation method
Choose your method by checking the bases. Matching bases let you compare exponents, provided the base is positive and not 1. If you cannot rewrite both sides with a shared base, apply logarithms and use their power rule. For powers of e, choose ln. Growth and decay models follow these methods too. Repeated doubling or halving tracks whole periods; for a target value, set the model equal to that target and solve. Keep full precision during calculations and round the final result.
Recovering an unknown exponent
Use logarithms to recover an exponent. For 3^x = 20, integer powers do not work. Taking logs gives x log 3 = log 20, so x ≈ 2.73. An equation containing e^(2x) and e^x may become a quadratic: set u = e^x, substitute and factor. Keep positive solutions for u, then find x = ln(u). For decay, divide elapsed time by the half-life. Each whole half-life halves the amount; fractional periods use the multiplier (1/2)^(time/half-life).
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