Using natural logarithms 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

Evaluating logarithmic expressions 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