Precalculus: 4.3 Rules of Loagrithms Concepts: rules of logarithms

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Precalculus: 4.3 Rules of Loagrithms

Concepts:rules of logarithms, change of base, solving equations.When working with polynomial, rational, and radical functions, the algebraic techniques we needed to be procient with

to perform manipulations on the functions were nding common denominator factoring long division of polynomials completing the square rationalizing numerator or denominator among others.

To perform manipulations on trigonometric functions, we need to be procient with trigonometric identities. That is why

trig identities are a big part of Math 1013 Precalculus II Trig.

To perform manipulations on exponential and logarithmic functions, we need to be procient with the rules of exponents

and the rules of logarithms. So these rules should be memorized, since they will form the basis of the techniques you will

use when working with exponential and logarithmic functions. The text takes the time to motivate where the rules come from. Laws of ExponentsIfxandyare real numbers, anda >0 is real, then

1.a0= 1

2.axay=ax+y

3. axa y=axy

4. (ax)y=axy

Laws of LogarithmsIfxandyare positive numbers, anda >0;b6= 1 is real, then

1. log

a(1) = 0

2. log

a(xy) = logax+ logay

3. log

axy = log axlogay

4. log

a(xr) =rlogaxwhere r is any real number

Inverse Function Cancellation

1. log

a(ax) =xfor everyx2(1;1)

2.aloga(x)=xfor everyx2(0;1)

In calculus, you will work most frequently with the natural logarithms, so I will also give you the rules with baseexand

lnxand suggest you memorize these rules and know how to change base to baseewhen necessary.

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Precalculus: 4.3 Rules of Loagrithms

Laws of ExponentsIfxandyare real numbers, then

1.e0= 1

2.exey=ex+y

3. exe y=exy

4. (ex)y=exy

Laws of LogarithmsIfxandyare positive numbers, then

1. ln(1) = 0

2. ln(xy) = lnx+ lny

3. ln xy = lnxlny

4. ln(xr) =rlnxwhere r is any real number

Inverse Function Cancellation

1. ln(ex) =x; x2(1;1)

2.elnx=x; x >0

Exponential Change of Base frombto basee

You can always convert to basee, using the following application of the rules: b x= (elnb)x =exlnb

Logarithm Change from Basebto basee

This requires a bit more work, but again uses the rules: y= logbx b y=blogbx b y=x ln(by) = ln(x) yln(b) = ln(x) y=ln(x)ln(b) This process can be used to change from any base to any other base.

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Precalculus: 4.3 Rules of Loagrithms

ExampleWrite log7xin terms of common and natural logarithms.

Convert to common logarithms:

Lety= log7x!7y=x.

7 y=x log(7 y) = logx ylog(7) = logx y=logxlog7 log

7x=logxlog7

Convert to natural logarithms:

Lety= log7x!7y=x.

7 y=x ln(7 y) = lnx yln(7) = lnx y=lnxln7 log

7x=lnxln7

ExampleSolvee53x= 10 forx.

Take the natural logarithm of both sides:

ln(e53x) = ln10

53x= ln10

3x= ln105

x=ln1053 x=5ln103

Example$1000 is deposited at 7.5% per year. If the interest paid is compounded daily, how long will it take for

the balance to reach $2000?

Solution

Lettbe the number of years after Jan 1.

P= $1000

A= $2000

r= 7:5% = 0:075 n= 365

The compound interest formula can be solved fort

A=P 1 +rn nt A=P= 1 +rn nt ln(A=P) = ln 1 +rn nt ln(A=P) =ntlnh 1 +rn i lnA=P)nln1 +rn =t t=ln(A=P)nln1 +rn ln(2000=1000)365ln[1 + 0:075=365]= 9:24291 It will take 9.24291 years for the accumulated amount to reach $2000.Page 3 of 3

Precalculus: 4.3 Rules of Loagrithms

Concepts:rules of logarithms, change of base, solving equations.When working with polynomial, rational, and radical functions, the algebraic techniques we needed to be procient with

to perform manipulations on the functions were nding common denominator factoring long division of polynomials completing the square rationalizing numerator or denominator among others.

To perform manipulations on trigonometric functions, we need to be procient with trigonometric identities. That is why

trig identities are a big part of Math 1013 Precalculus II Trig.

To perform manipulations on exponential and logarithmic functions, we need to be procient with the rules of exponents

and the rules of logarithms. So these rules should be memorized, since they will form the basis of the techniques you will

use when working with exponential and logarithmic functions. The text takes the time to motivate where the rules come from. Laws of ExponentsIfxandyare real numbers, anda >0 is real, then

1.a0= 1

2.axay=ax+y

3. axa y=axy

4. (ax)y=axy

Laws of LogarithmsIfxandyare positive numbers, anda >0;b6= 1 is real, then

1. log

a(1) = 0

2. log

a(xy) = logax+ logay

3. log

axy = log axlogay

4. log

a(xr) =rlogaxwhere r is any real number

Inverse Function Cancellation

1. log

a(ax) =xfor everyx2(1;1)

2.aloga(x)=xfor everyx2(0;1)

In calculus, you will work most frequently with the natural logarithms, so I will also give you the rules with baseexand

lnxand suggest you memorize these rules and know how to change base to baseewhen necessary.

Page 1 of 3

Precalculus: 4.3 Rules of Loagrithms

Laws of ExponentsIfxandyare real numbers, then

1.e0= 1

2.exey=ex+y

3. exe y=exy

4. (ex)y=exy

Laws of LogarithmsIfxandyare positive numbers, then

1. ln(1) = 0

2. ln(xy) = lnx+ lny

3. ln xy = lnxlny

4. ln(xr) =rlnxwhere r is any real number

Inverse Function Cancellation

1. ln(ex) =x; x2(1;1)

2.elnx=x; x >0

Exponential Change of Base frombto basee

You can always convert to basee, using the following application of the rules: b x= (elnb)x =exlnb

Logarithm Change from Basebto basee

This requires a bit more work, but again uses the rules: y= logbx b y=blogbx b y=x ln(by) = ln(x) yln(b) = ln(x) y=ln(x)ln(b) This process can be used to change from any base to any other base.

Page 2 of 3

Precalculus: 4.3 Rules of Loagrithms

ExampleWrite log7xin terms of common and natural logarithms.

Convert to common logarithms:

Lety= log7x!7y=x.

7 y=x log(7 y) = logx ylog(7) = logx y=logxlog7 log

7x=logxlog7

Convert to natural logarithms:

Lety= log7x!7y=x.

7 y=x ln(7 y) = lnx yln(7) = lnx y=lnxln7 log

7x=lnxln7

ExampleSolvee53x= 10 forx.

Take the natural logarithm of both sides:

ln(e53x) = ln10

53x= ln10

3x= ln105

x=ln1053 x=5ln103

Example$1000 is deposited at 7.5% per year. If the interest paid is compounded daily, how long will it take for

the balance to reach $2000?

Solution

Lettbe the number of years after Jan 1.

P= $1000

A= $2000

r= 7:5% = 0:075 n= 365

The compound interest formula can be solved fort

A=P 1 +rn nt A=P= 1 +rn nt ln(A=P) = ln 1 +rn nt ln(A=P) =ntlnh 1 +rn i lnA=P)nln1 +rn =t t=ln(A=P)nln1 +rn ln(2000=1000)365ln[1 + 0:075=365]= 9:24291 It will take 9.24291 years for the accumulated amount to reach $2000.Page 3 of 3