502 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions y = 353ekt Solve for k. 2000 = 353ek12852 Substitute 2000 for y and 285 for t. 2000 353 = e285k Divide by 353. ln 2000 353 = ln e285k Take the natural logarithm on each side. ln 2000 353 = 285k ln ex = x, for all x. k = 1 285 # ln 2000 353 Multiply by 1 285 and rewrite. k ≈0.00609 Use a calculator. A function that models the data is y = 353e0.00609t. (b) y = 353e0.00609t Solve the model from part (a) for the year t. 560 = 353e0.00609t To double the level 280, let y = 212802 = 560. 560 353 = e0.00609t Divide by 353. ln 560 353 = ln e0.00609t Take the natural logarithm on each side. ln 560 353 = 0.00609t ln ex = x, for all x. t = 1 0.00609 # ln 560 353 Multiply by 1 0.00609 and rewrite. t ≈75.8 Use a calculator. Since t = 0 corresponds to 1990, the preindustrial carbon dioxide level will double in the 75th year after 1990, or during 2065, according to this model. S Now Try Exercise 43. EXAMPLE 2 Finding DoublingTime for Money How long, to the nearest year, will it take for money in an account that accrues interest at a rate of 3%, compounded continuously, to double? SOLUTION A = Per t Continuous compounding formula 2P = Pe0.03t Let A = 2P and r = 0.03. 2 = e0.03t Divide by P. ln 2 = ln e0.03t Take the natural logarithm on each side. ln 2 = 0.03t ln ex = x ln 2 0.03 = t Divide by 0.03. 23.10 ≈t Use a calculator. It will take about 23 yr for the amount to double. S Now Try Exercise 31.
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