Solve the Equation

Solve the following equation. Write the answer in terms of the natural logarithm.

e2x=9

Question

Solve the Equation

Solve the following equation. Write the answer in terms of the natural logarithm.

e2x=9

BlackTom AI · Accepted Answer

The problem asks to solve the equation and express the solution in terms of the natural logarithm.
Option 1: ln(9)/2. This comes from taking natural logs on both sides of e^{2x} = 9, yielding 2x = ln(9), and then dividing by 2 to isolate x, giving x = (ln 9)/2.
Option 2: ln(3). Since 9 = 3^2, we can use the log property ln(9) = ln(3^2) = 2 ln(3). Substituting into (ln 9)/2 gives (2 ln(3))/2 = ln(3). So this is an algebraic simplification of the first expression.
Why the two expressions are related: the logarithm rules allow moving constants: if you have (ln(9))/2, you can rewrite ln(9) as 2 ln(3), and the 2 factors cancel, yielding ln(3). Therefore, both forms represent the same numerical value for x.
What would make an option incorrect: if an option attempted to discard the log form entirely or used a logarithm base other than e, that would not align with the instruction to express in natural logarithm terms. Also, if an option claimed a different numeric result that contradicts ln(9) = 2 ln(3), it would be incorrect. In this case, the core idea is applying the inverse relationship between the exponential and natural logarithm, then simplifying using logarithm identities.

MAT-265-OD24-02-7799

Solve the Equation Solve the following equation. Write the answer in terms of the natural logarithm. e2x=9

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