L> Logarithms

LOGARITHMSThe inverse of the exponential function-- The Natural LogarithmThe graph of y = ex

clearly shows that it is a one to lớn one function, hence an inverseexists. We điện thoại tư vấn this inverse the naturallogarithm. And write it as

y = ln x

Below is the graph of

y = ex andy = ln xBy definition, they are reflections of each other across theline y = x.

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Inverse Properties of LogsSince logs & exponents cancel each other we have:eln x = xandln ex = xExampleeln 3 =3andln(e5) = 5

Three Properties of Logs Property 1: ln (uv) = ln u + ln v (The product to Sum Rule) Property 2: ln (u/v) = ln u - ln v (The Quotient to lớn Difference Rule) Property 3: ln ur = r ln u (The power nguồn Rule) Example:Expand: ln (xy2/z)by property 2 we have:ln (xy2) - ln zby property 1 we haveln x + ln y2 - ln zBy property 3 we haveln x + 2 ln y- ln zExerciseTry to expand

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ExampleWrite as a single logarithm:4ln x - 1/2ln y + ln zSolutionWe first use property 3 khổng lồ write:ln x4 - ln y1/2 + ln zNow we use property 2:x4 ln+ ln z y1/2Finally, we use property 3:x4 z ln+ ln z y1/2ExerciseWrite the following as a single logarithm:1/3 ln x + 2 ln y - 3 ln z" width="101" height="22">ExampleSuppose that ln 3 = 1.10and thatln 5 = 1.61Findln 45SolutionSince45 = (5)(32)We haveln 45 = ln (5)(32) = ln 5 + ln 32= 1.61 +2 ln 3= 1.61 + 2 (1.10) = 3.81 Exponential Equations

The key lớn solving equations is khổng lồ know how khổng lồ apply the inverseof a function. When we have an exponential equation, we will use thenatural logarithm khổng lồ cancel the exponential.

Example

Find k if

34 = e10k

Solution

Take the natural logarithm of both sides

ln 34 = ln e10k

Now use the inverse property

ln 34 = 10k

Finally divide by 10

ln 34= k = 0.3526 10

Carbon Dating

All living beings have a certain amount of radioactive carbon C14 in their bodies. When the being dies the C14 slowly decays with a half life of about 5600 years. Suppose a skeleton is found in Tahoe that has 42% of the original C14.


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When did the person die? Solution We can use the exponential decay equation: y = Cekt After 5600 years there is C/2 C14 left. Substituting, we get: C/2 = Cek(5600) Dividing by C, một nửa = e5600k Take ln of both sides, ln(0.5) = 5600k so that ln(0.5) k = = -0.000124 5600 The equation becomes y = Ce-0.000124t khổng lồ find out when the person died, substitute y = 0.42C và solve for t: 0.42C = Ce-0.000124t Divide by C, 0.42 = e-0.000124t Take ln of both sides, ln(0.42) = -0.000124t Divide by -0.000124 ln(0.42) t = = 6995 -0.000124 The person died about 7,000 years ago.Backto the Math 116 trang chủ PageBack lớn the MathDepartment Home