Solve for x: e^(2x) = 7. Express your answer using natural logarithm.

Question

Accepted Answer

ln(7) / 2. Starting with e^(2x) = 7, take the natural logarithm of both sides: ln(e^(2x)) = ln(7). Since ln and e are inverse operations, the left side simplifies to 2x: 2x = ln(7). Dividing both sides by 2: x = ln(7)/2.

This is approximately x = 1.9459/2 = 0.9730, but the exact answer ln(7)/2 is preferred on the CLEP exam. "2 ln(7)" (2 ln(7)) would be the answer to e^x = 7^2 = 49. "ln(7/2)" (ln(7/2)) would equal ln(3.5) = 1.253, which is incorrect. "ln(14)" (ln(14)) equals ln(7) + ln(2), not ln(7)/2. Be careful to distinguish between dividing the logarithm by 2 and dividing the argument by 2.

Solving exponential equations using logarithms is a core CLEP skill. When the base is e, use natural logarithm (ln). When the base is 10, use common logarithm (log). For other bases, you can use either ln or log with the change-of-base formula: log_b(x) = ln(x)/ln(b) = log(x)/log(b). The key algebraic step is always isolating the exponential expression first, then applying the appropriate logarithm to both sides.

CLEP College Algebra: Question of the Day

Get a free CLEP College Algebra question every day

Ready for the full CLEP College Algebra? Earn 3 college credits.