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Lesson 81. Using natural logarithms. Natural logarithms. When the base is e , the logarithm is called a natural logarithm e x = a means ln a= x. Inverse properties of logarithms. e ln x = x and ln e x = x where x>0.
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Lesson 81 Using natural logarithms
Natural logarithms • When the base is e, the logarithm is called a natural logarithm • ex = a means ln a= x
Inverse properties of logarithms • eln x = x and ln ex = x • where x>0
Simplifying exponential and logarithmic expressions • Simplify elnp • elnp = p • Simplify ln e3c2+1= • = 3c2+1 • simplfy eln2x • simplify ln e2d2+d
Properties of natural logarithms • Product property • ln ab = ln a + ln b • Quotient Property • ln (a/b) = ln a - ln b • Power Property • ln ap = p ln a
Applying properties of natural logarithms • Rewrite as a sum or difference • ln 6e= ln 6 + ln e • = ln 6 + 1 • ln 3e = ln 3e- ln x=ln3 + ln e-ln x • x • =ln 3 +1 - lnx • ln e5x2 = 5x2 ln e = 5x2 (1) = 5x2
practice • Write as a sum or difference- then simplify • ln 10e • ln 5e • y • 3 ln e6x2 • ln( 4c3)4 • e2
Continuous exponential growth • The formula for exponential growth where interest is compounded continuously is • A = P ert
Lesson 87 • Evaluating logarithmic expressions
Evaluating expressions of the form loga(bc)d • Use the properties of logs to evaluate • log4(16x)3 when x = 256 • =3log4(16x) • =3(log416+log4x) • Since log416 = 2 • = 3(2+log4x) • = 6 +3log4x • When x = 256 • = 6+ 3log4 256= 6 + 3 (4)= 18
evaluate • ln(7e)2 • = 2 ln(7e) • =2 (ln 7+ln e) • = 2(ln 7 + 1) • = 2 ln 7 + 2 • = 2(1.95) + 2 = 5.9
practice • Evaluate when r = 243 • log3(27r)4 • Evaluate • ln(12e)3
the change of base formula • remember the change of base formula where a is the new base
using the change of base formula • Convert log100(10x)2 to base 10. then evaluate when x = 1000 • =2 log100 (10x) • =2( log 10x) • log 100 • = 2 (log10 +logx) • log100 • =2( 1 + log 1000)= 2( 1+3) = 4 • 2 2
use change of base formula • Convert to base e, then evaluate when x = 6 • log4(2x)3= 3 log4(2x) • =3 (ln 2x)= 3 (ln 2 + ln x) • ln 4 ln 4 • = 3 ( ln2 + ln 6) • ln 4 • = 3( .6931 + 1.7918) = 5.3774 • 1.3863
practice • solve - change to base 10 • log 100(1000x)3 when x = 10 • change to base e • log9(3x)5 when x = 4
Solving log equations using the change of base formula • Solve 1000 9x = 100 • log 1000 100 = 9x • change to base 10 • log 100 = 9x • log 1000 • 2 = 9x • 3 • 2 = 27x x = 2/27
practice • Solve for x: • 32 4x = 8