1Cademy - Try It 10.83 and 10.84: Solving $$2e^{x-2} = 18$$ and $$5e^{2x} = 25$$

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Exponential Equation

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Try It 10.83 and 10.84: Solving $2e^{x-2} = 18$ and $5e^{2x} = 25$

Practice solving exponential equations with base $e$ by using the natural logarithm. For the equation $2e^{x-2} = 18$ , first isolate the exponential by dividing by $2$ to get $e^{x-2} = 9$ . Take the natural logarithm of both sides, resulting in $\ln e^{x-2} = \ln 9$ . Using the Power Property and knowing $\ln e = 1$ , this simplifies to $x-2 = \ln 9$ . Adding $2$ gives the exact solution $x = \ln 9 + 2$ , approximately $4.197$ . For the equation $5e^{2x} = 25$ , divide by $5$ to isolate the exponential, yielding $e^{2x} = 5$ . Take the natural logarithm of both sides to get $\ln e^{2x} = \ln 5$ , which simplifies to $2x = \ln 5$ . Dividing by $2$ gives the exact solution $x = \frac{\ln 5}{2}$ , approximately $0.805$ .

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Updated 2026-05-25

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OpenStax Intermediate Algebra

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