Math 171 Group Worksheet over Exponentials/Logarithms (1.6) January 19, 2016 Credit given for work shown. No calculators needed. Name √ 1. Calculate ln[e2/3 e5 ] 2. Write as the natural log of a single expression. 2 ln(x1/3 ) + ln(7x) 2 ln(3) + 4 ln(2) 3. Solve for the unknown variable. [Note: You can do these without logs.] √ 1 ( 125)x = 25 w3/2 = 64 36x = 216 4. Suppose A = log x and B = log y. Using properties of logarithms, find equivalent expressions in terms of A and/or B. Your final answers should NOT involve any logs. log(y/x) = log(10/y 2 ) = log(100xy) = ex − e−x ex + e−x and cosh x = , simplify cosh2 x + sinh2 x. [Your answer should 2 2 not contain e.] 5. If sinh x = 6. One hundred grams of a particular radioactive substance decays according to the function m(t) = 100e−t/650 , where t > 0 measures times in years. When does the mass reach 25 grams? 7. In 1935 Charles Richter defined the magnitude of an earthquake to be M = log I S where I is the intensity of the earthquake (measured by the amplitude of a seismograph reading taken 100 km from the epicenter of the earthquake) and S is the intensity of a ”standard earthquake” (whose amplitude is 1 micron = 10−4 cm). Early in the century the earthquake in San Francisco registered 8.3 on the Richter scale. In the same year, another earthquake was recorded in South America that was four times stronger. What was the magnitude of the earthquake in South American? 8. When a camera flash goes off, the batteries immediately begin to recharge the flash’s capacitor, which stores electrical charge given by Q(t) = Q0 (1 − e−t/a ) where t is measured in seconds, and Q0 and c > 0 are physical constants. The steady-state charge is the value that Q(t) approaches as t becomes large. What is the steady-state charge?