Integration that leads to logarithm functions









Properties of Exponents and Logarithms

Therefore lnx = y if and only if ey = x. Most calculators can directly compute logs base 10 and the natural log. For any other base it is necessary to use 
Exponents and Logarithms


Limits involving ln(x)

We can use the rules of logarithms given above to derive the following information about limits. lim x→∞ ln x = ∞ lim x→0.
. Limits Derivatives and Integrals


Significant Figure Rules for logs

Regular sig fig rules are guidelines and they don't always predict the correct The rule for natural logs (ln) is similar
Significant Figure Rules for logs


Logarithms

state and use the laws of logarithms. • solve simple equations requiring the use of logarithms. Contents. 1. Introduction log and ln.
mc ty logarithms





The laws of logarithms

a) 3 log10 5 b) 2 log x
mc bus loglaws


6.2 Properties of Logarithms

3. ln. ( 3 ex. )2. 4. log 3. √. 100x2 yz5. 5. log117(x2 − 4). Solution. 1. To expand log2. (8 x) we use the Quotient Rule identifying u = 8 and w = x and 
S&Z . & .


What is a logarithm? Log base 10

And so ln(ex) = x eln(x) = x. • Now we have a new set of rules to add to the others: Table 4. Functions of log base 10 and base e. Exponents. Log base 10.
logarithms


Logarithms Tutorial for Chemistry Students 1 Logarithms

There are four main algebraic rules used to manipulate logarithms: natural logarithms we can employ the following simple identity: ln(x)=2.303 log(x).
LogarithmsTutorial





Download Free Natural Logarithm Examples And Answers

5 days ago The 11 Natural Log Rules You Need to Know. Logarithm Questions and Answers ... Solving Natural Log Equations Solving a natural logarithmic.


Integration that leads to logarithm functions

ln x + c. In this unit we generalise this result and see how a wide variety of integrals result in logarithm functions. In order to master the techniques 
mc ty inttologs


213088 Integration that leads to logarithm functions

Integration that leadsto logarithm functions

mc-TY-inttologs-2009-1 The derivative oflnxis1x. As a consequence, if we reverse the process, the integral of1xis lnx+c. In this unit we generalise this result and see how a wide variety of integrals result in logarithm functions. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: •recognise integrals in which the numerator is the derivative of the denominator. •rewrite integrals in alternative forms so that the numerator becomes the derivative of the denominator. •recognise integrals which can lead to logarithm functions.

Contents

1.Introduction 2

2.Some examples 3

www.mathcentre.ac.uk 1c?mathcentre 2009

1. IntroductionWe already know that when we differentiatey= lnxwe finddy

dx=1x. We also know that if we havey= lnf(x)and we differentiate it we finddy dx=f?(x)f(x). The point is that if we recognise that the function we are trying to integrate is the derivative of another function, we can simply reverse the process. So if the function we are trying to integrate is a quotient, and if the numerator is the derivative of the denominator, then the integral will involve a logarithm: ify= lnf(x)so thatdy dx=f?(x)f(x) and, reversing the process, ?f?(x) f(x)dx= ln(f(x)) +c. This procedure works if the functionf(x)is positive, because then we can take its logarithm. What happens if the function is negative? In that case,-f(x)is positive, so that we can take the logarithm of-f(x). Then: ify= ln(-f(x))so thatdy dx=-f?(x)-f(x)=f?(x)f(x) and, reversing the process, ?f?(x) f(x)dx= ln(-f(x)) +c when the function is negative. We can combine both these results by using the modulus function. Then we can use the formula in both cases, or when the function takes both positive and negative values (or when we don"t know).

Key Point

To integrate a quotient when the numerator is the derivativeof the denominator, we use ?f?(x) f(x)dx= ln|f(x)|+c. www.mathcentre.ac.uk 2c?mathcentre 2009

2. Some examplesExampleFind?

tanxdx.

Recall that we can rewritetanxassinx

cosx. Observe that the derivative ofcosxis-sinx, so that the numerator is very nearly the derivative of the denominator. We make it so by rewritingsinx cosx as--sinx cosxand the integral becomes tanxdx=?sinx cosxdx =-?-sinx cosxdx =-ln|cosx|+c.

Integration that leadsto logarithm functions

mc-TY-inttologs-2009-1 The derivative oflnxis1x. As a consequence, if we reverse the process, the integral of1xis lnx+c. In this unit we generalise this result and see how a wide variety of integrals result in logarithm functions. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: •recognise integrals in which the numerator is the derivative of the denominator. •rewrite integrals in alternative forms so that the numerator becomes the derivative of the denominator. •recognise integrals which can lead to logarithm functions.

Contents

1.Introduction 2

2.Some examples 3

www.mathcentre.ac.uk 1c?mathcentre 2009

1. IntroductionWe already know that when we differentiatey= lnxwe finddy

dx=1x. We also know that if we havey= lnf(x)and we differentiate it we finddy dx=f?(x)f(x). The point is that if we recognise that the function we are trying to integrate is the derivative of another function, we can simply reverse the process. So if the function we are trying to integrate is a quotient, and if the numerator is the derivative of the denominator, then the integral will involve a logarithm: ify= lnf(x)so thatdy dx=f?(x)f(x) and, reversing the process, ?f?(x) f(x)dx= ln(f(x)) +c. This procedure works if the functionf(x)is positive, because then we can take its logarithm. What happens if the function is negative? In that case,-f(x)is positive, so that we can take the logarithm of-f(x). Then: ify= ln(-f(x))so thatdy dx=-f?(x)-f(x)=f?(x)f(x) and, reversing the process, ?f?(x) f(x)dx= ln(-f(x)) +c when the function is negative. We can combine both these results by using the modulus function. Then we can use the formula in both cases, or when the function takes both positive and negative values (or when we don"t know).

Key Point

To integrate a quotient when the numerator is the derivativeof the denominator, we use ?f?(x) f(x)dx= ln|f(x)|+c. www.mathcentre.ac.uk 2c?mathcentre 2009

2. Some examplesExampleFind?

tanxdx.

Recall that we can rewritetanxassinx

cosx. Observe that the derivative ofcosxis-sinx, so that the numerator is very nearly the derivative of the denominator. We make it so by rewritingsinx cosx as--sinx cosxand the integral becomes tanxdx=?sinx cosxdx =-?-sinx cosxdx =-ln|cosx|+c.
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