Learning Objectives
By the end of this section, you will be able to:
Use the properties of logarithms
Use the Change of Base Formula
Be Prepared 10.10
Before you get started, take this readiness quiz.
Evaluate: ⓐ a0a0 ⓑ a1.a1. If you missed this problem, review Example 5.14.
Be Prepared 10.11
Write with a rational exponent: x2y3.x2y3. If you missed this problem, review Example 8.27.
Be Prepared 10.12
Round to three decimal places: 2.5646415. If you missed this problem, review Example 1.34.
Use the Properties of Logarithms
Now that we have learned about exponential and logarithmic functions, we can introduce some of the properties of logarithms. These will be very helpful as we continue to solve both exponential and logarithmic equations.
The first two properties derive from the definition of logarithms. Since a0=1,a0=1, we can convert this to logarithmic form and get loga1=0.loga1=0. Also, since a1=a,a1=a, we get logaa=1.logaa=1.
Properties of Logarithms
In the next example we could evaluate the logarithm by converting to exponential form, as we have done previously, but recognizing and then applying the properties saves time.
Example 10.28
Evaluate using the properties of logarithms: ⓐ log81log81 and ⓑ log66.log66.
Solution
ⓐ
row: log81log81
row: Use the property, loga1=0loga1=0. | 0log81=00log81=0
ⓑ log66Use the property,logaa=1.1log66=1log66Use the property,logaa=1.1log66=1
Try It 10.55
Evaluate using the properties of logarithms: ⓐ log131log131 ⓑ log99.log99.
Try It 10.56
Evaluate using the properties of logarithms: ⓐ log51log51 ⓑ log77.log77.
The next two properties can also be verified by converting them from exponential form to logarithmic form, or the reverse.
The exponential equation alogax=xalogax=x converts to the logarithmic equation logax=logax,logax=logax, which is a true statement for positive values for x only.
The logarithmic equation logaax=xlogaax=x converts to the exponential equation ax=ax,ax=ax, which is also a true statement.
These two properties are called inverse properties because, when we have the same base, raising to a power “undoes” the log and taking the log “undoes” raising to a power. These two properties show the composition of functions. Both ended up with the identity function which shows again that the exponential and logarithmic functions are inverse functions.
Inverse Properties of Logarithms
For a>0,a>0,x>0x>0 and a≠1,a≠1,
In the next example, apply the inverse properties of logarithms.
Example 10.29
Evaluate using the properties of logarithms: ⓐ 4log494log49 and ⓑ log335.log335.
Solution
ⓐ
row: 4log494log49
row: Use the property, alogax=xalogax=x. | 94log49=994log49=9
ⓑ
row: log335log335
row: Use the property, alogax=xalogax=x. | 5log335=55log335=5
Try It 10.57
Evaluate using the properties of logarithms: ⓐ 5log5155log515 ⓑ log774.log774.
Try It 10.58
Evaluate using the properties of logarithms: ⓐ 2log282log28 ⓑ log2215.log2215.
There are three more properties of logarithms that will be useful in our work. We know exponential functions and logarithmic function are very interrelated. Our definition of logarithm shows us that a logarithm is the exponent of the equivalent exponential. The properties of exponents have related properties for exponents.
In the Product Property of Exponents, am·an=am+n,am·an=am+n, we see that to multiply the same base, we add the exponents. The Product Property of Logarithms, logaM·N=logaM+logaNlogaM·N=logaM+logaN tells us to take the log of a product, we add the log of the factors.
Product Property of Logarithms
If M>0,N>0,a>0M>0,N>0,a>0 and a≠1,a≠1, then,
The logarithm of a product is the sum of the logarithms.
We use this property to write the log of a product as a sum of the logs of each factor.
Example 10.30
Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible: ⓐ log37xlog37x and ⓑ log464xy.log464xy.
Solution
ⓐ
row: log37xlog37x
row: Use the Product Property, loga(M·N)=logaM+logaNloga(M·N)=logaM+logaN. | log37+log3xlog37+log3x
row: log37x=log37+log3xlog37x=log37+log3x
ⓑ
row: log464xylog464xy
row: Use the Product Property, loga(M·N)=logaM+logaNloga(M·N)=logaM+logaN. | log464+log4x+log4ylog464+log4x+log4y
row: Simplify by evaluating log464log464. | 3+log4x+log4y3+log4x+log4y
row: log464xy=3+log4x+log4ylog464xy=3+log4x+log4y
Try It 10.59
Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.
ⓐ log33xlog33x ⓑ log28xylog28xy
Try It 10.60
Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.
ⓐ log99xlog99x ⓑ log327xylog327xy
Similarly, in the Quotient Property of Exponents, aman=am−n,aman=am−n, we see that to divide the same base, we subtract the exponents. The Quotient Property of Logarithms, logaMN=logaM−logaNlogaMN=logaM−logaN tells us to take the log of a quotient, we subtract the log of the numerator and denominator.
Quotient Property of Logarithms
If M>0,N>0,a>0M>0,N>0,a>0 and a≠1,a≠1, then,
The logarithm of a quotient is the difference of the logarithms.
Note that logaM−logaN≠loga(M−N).logaM−logaN≠loga(M−N).
We use this property to write the log of a quotient as a difference of the logs of each factor.
Example 10.31
Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.ⓐ log557log557 and ⓑ logx100logx100
Solution
ⓐ
row: log557log557
row: Use the Quotient Property, logaMN=logaM−logaNlogaMN=logaM−logaN. | log55−log57log55−log57
row: Simplify. | 1−log571−log57
row: log557=1−log57log557=1−log57
ⓑ
row: logx100logx100
row: Use the Quotient Property, logaMN=logaM−logaNlogaMN=logaM−logaN. | logx−log100logx−log100
row: Simplify. | logx−2logx−2
row: logx100=logx−2logx100=logx−2
Try It 10.61
Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.
ⓐ log434log434 ⓑ logx1000logx1000
Try It 10.62
Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.
ⓐ log254log254 ⓑ log10ylog10y
The third property of logarithms is related to the Power Property of Exponents, (am)n=am·n,(am)n=am·n, we see that to raise a power to a power, we multiply the exponents. The Power Property of Logarithms, logaMp=plogaMlogaMp=plogaM tells us to take the log of a number raised to a power, we multiply the power times the log of the number.
Power Property of Logarithms
If M>0,a>0,a≠1M>0,a>0,a≠1 and pp is any real number then,
The log of a number raised to a power is the product of the power times the log of the number.
We use this property to write the log of a number raised to a power as the product of the power times the log of the number. We essentially take the exponent and throw it in front of the logarithm.
Example 10.32
Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.ⓐ log543log543 and ⓑ logx10logx10
Solution
ⓐ
row: log543log543
row: Use the Power Property, logaMp=plogaMlogaMp=plogaM. | 3log543log54
row: log543=3log54log543=3log54
ⓑ
row: logx10logx10
row: Use the Power Property, logaMp=plogaMlogaMp=plogaM. | 10logx10logx
row: logx10=10logxlogx10=10logx
Try It 10.63
Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.
ⓐ log754log754 ⓑ logx100logx100
Try It 10.64
Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.
ⓐ log237log237 ⓑ logx20logx20
We summarize the Properties of Logarithms here for easy reference. While the natural logarithms are a special case of these properties, it is often helpful to also show the natural logarithm version of each property.
Properties of Logarithms
If M>0,N>0,a>0,a≠1M>0,N>0,a>0,a≠1 and pp is any real number then,
row: Property | Base aa | Base ee
row: loga1=0loga1=0 | ln1=0ln1=0
row: logaa=1logaa=1 | lne=1lne=1
row: Inverse Properties | alogax=xlogaax=xalogax=xlogaax=x | elnx=x lnex=xelnx=x lnex=x
row: Product Property of Logarithms | loga(M·N)=logaM+logaNloga(M·N)=logaM+logaN | ln(M·N)=lnM+lnNln(M·N)=lnM+lnN
row: Quotient Property of Logarithms | logaMN=logaM−logaNlogaMN=logaM−logaN | lnMN=lnM−lnNlnMN=lnM−lnN
row: Power Property of Logarithms | logaMp=plogaMlogaMp=plogaM | lnMp=plnMlnMp=plnM
Now that we have the properties we can use them to “expand” a logarithmic expression. This means to write the logarithm as a sum or difference and without any powers.
We generally apply the Product and Quotient Properties before we apply the Power Property.
Example 10.33
Use the Properties of Logarithms to expand the logarithm log4(2x3y2)log4(2x3y2). Simplify, if possible.
Solution
row: log4(2x3y2)log4(2x3y2)
row: Use the Product Property, logaM·N=logaM+logaNlogaM·N=logaM+logaN. | log42+log4x3+log4y2log42+log4x3+log4y2
row: Use the Power Property, logaMp=plogaMlogaMp=plogaM, on the last two terms. | log42+3log4x+2log4ylog42+3log4x+2log4y
row: Simplify. | 12+3log4x+2log4y12+3log4x+2log4y
row: log4(2x3y2)=12+3log4x+2log4ylog4(2x3y2)=12+3log4x+2log4y
Try It 10.65
Use the Properties of Logarithms to expand the logarithm log2(5x4y2)log2(5x4y2). Simplify, if possible.
Try It 10.66
Use the Properties of Logarithms to expand the logarithm log3(7x5y3)log3(7x5y3). Simplify, if possible.
When we have a radical in the logarithmic expression, it is helpful to first write its radicand as a rational exponent.
Example 10.34
Use the Properties of Logarithms to expand the logarithm log2x33y2z4log2x33y2z4. Simplify, if possible.
Solution
row: log2x33y2z4log2x33y2z4
row: Rewrite the radical with a rational exponent. | log2(x33y2z)14log2(x33y2z)14
row: Use the Power Property, logaMp=plogaMlogaMp=plogaM. | 14log2(x33y2z)14log2(x33y2z)
row: Use the Quotient Property, logaM·N=logaM−logaNlogaM·N=logaM−logaN. | 14(log2(x3)−log2(3y2z))14(log2(x3)−log2(3y2z))
row: Use the Product Property, logaM·N=logaM+logaNlogaM·N=logaM+logaN, in the second term. | 14(log2(x3)−(log23+log2y2+log2z))14(log2(x3)−(log23+log2y2+log2z))
row: Use the Power Property, logaMp=plogaMlogaMp=plogaM, inside the parentheses. | 14(3log2x−(log23+2log2y+log2z))14(3log2x−(log23+2log2y+log2z))
row: Simplify by distributing. | 14(3log2x−log23−2log2y−log2z)14(3log2x−log23−2log2y−log2z)
row: log2x33y2z4=14(3log2x−log23−2log2y−log2z)log2x33y2z4=14(3log2x−log23−2log2y−log2z)
Try It 10.67
Use the Properties of Logarithms to expand the logarithm log4x42y3z25log4x42y3z25. Simplify, if possible.
Try It 10.68
Use the Properties of Logarithms to expand the logarithm log3x25yz3log3x25yz3. Simplify, if possible.
The opposite of expanding a logarithm is to condense a sum or difference of logarithms that have the same base into a single logarithm. We again use the properties of logarithms to help us, but in reverse.
To condense logarithmic expressions with the same base into one logarithm, we start by using the Power Property to get the coefficients of the log terms to be one and then the Product and Quotient Properties as needed.
Example 10.35
Use the Properties of Logarithms to condense the logarithm log43+log4x−log4ylog43+log4x−log4y. Simplify, if possible.
Solution
row: The log expressions all have the same base, 4. | log43+log4x−log4ylog43+log4x−log4y
row: The first two terms are added, so we use the Product Property, logaM+logaN=logaM·NlogaM+logaN=logaM·N. | log43x−log4ylog43x−log4y
row: Since the logs are subtracted, we use the Quotient Property, logaM−logaN=logaMNlogaM−logaN=logaMN. | log43xylog43xy
row: log43+log4x−log4y=log43xylog43+log4x−log4y=log43xy
Try It 10.69
Use the Properties of Logarithms to condense the logarithm log25+log2x−log2ylog25+log2x−log2y. Simplify, if possible.
Try It 10.70
Use the Properties of Logarithms to condense the logarithm log36−log3x−log3ylog36−log3x−log3y. Simplify, if possible.
Example 10.36
Use the Properties of Logarithms to condense the logarithm 2log3x+4log3(x+1)2log3x+4log3(x+1). Simplify, if possible.
Solution
row: The log expressions have the same base, 3. | 2log3x+4log3(x+1)2log3x+4log3(x+1)
row: Use the Power Property, logaM+logaN=logaM·NlogaM+logaN=logaM·N. | log3x2+log3(x+1)4log3x2+log3(x+1)4
row: The terms are added, so we use the Product Property, logaM+logaN=logaM·NlogaM+logaN=logaM·N. | log3x2(x+1)4log3x2(x+1)4
row: 2log3x+4log3(x+1)=log3x2(x+1)42log3x+4log3(x+1)=log3x2(x+1)4
Try It 10.71
Use the Properties of Logarithms to condense the logarithm 3log2x+2log2(x−1)3log2x+2log2(x−1). Simplify, if possible.
Try It 10.72
Use the Properties of Logarithms to condense the logarithm 2logx+2log(x+1)2logx+2log(x+1). Simplify, if possible.
Use the Change-of-Base Formula
To evaluate a logarithm with any other base, we can use the Change-of-Base Formula. We will show how this is derived.
row: Suppose we want to evaluate logaMlogaM. | logaMlogaM
row: Let y=logaMy=logaM. | y=logaMy=logaM
row: Rewrite the expression in exponential form. | ay=May=M
row: Take the logblogb of each side. | logbay=logbMlogbay=logbM
row: Use the Power Property. | ylogba=logbMylogba=logbM
row: Solve for yy. | y=logbMlogbay=logbMlogba
row: Substitute y=logaMy=logaM. | logaM=logbMlogbalogaM=logbMlogba
The Change-of-Base Formula introduces a new base b.b. This can be any base b we want where b>0,b≠1.b>0,b≠1. Because our calculators have keys for logarithms base 10 and base e, we will rewrite the Change-of-Base Formula with the new base as 10 or e.
ਬੇਸ ਬਦਲਣ ਦਾ ਸੂਤਰ
ਕਿਸੇ ਵੀ ਲਘੂਗਣਕ ਬੇਸ a, b ਅਤੇ M>0 ਲਈ,
ਜਦੋਂ ਅਸੀਂ ਲਘੂਗਣਕ ਮੁੱਲ ਪਤਾ ਕਰਨ ਲਈ ਕੈਲਕੁਲੇਟਰ ਦੀ ਵਰਤੋਂ ਕਰਦੇ ਹਾਂ, ਅਸੀਂ ਆਮ ਤੌਰ 'ਤੇ ਤਿੰਨ ਦਸ਼ਮਲਵ ਸਥਾਨਾਂ ਤੱਕ ਗੋਲ ਕਰਦੇ ਹਾਂ। ਇਹ ਸਾਨੂੰ ਇੱਕ ਲਗਭਗ ਮੁੱਲ ਦਿੰਦਾ ਹੈ ਅਤੇ ਇਸ ਲਈ ਅਸੀਂ ਲਗਭਗ ਬਰਾਬਰ ਚਿੰਨ੍ਹ (≈) ਦੀ ਵਰਤੋਂ ਕਰਦੇ ਹਾਂ।
ਉਦਾਹਰਨ 10.37
ਤਿੰਨ ਦਸ਼ਮਲਵ ਸਥਾਨਾਂ ਤੱਕ ਗੋਲ ਕਰਕੇ, ਲਗਭਗ log435।
ਹੱਲ
ਸਤਰ: ਬੇਸ ਬਦਲਣ ਦੇ ਸੂਤਰ ਦੀ ਵਰਤੋਂ ਕਰੋ।
ਸਤਰ: a ਅਤੇ M ਦੀ ਪਛਾਣ ਕਰੋ। b ਲਈ 10 ਚੁਣੋ।
ਸਤਰ: ਬੇਸ 10 ਲਈ ਲਾਗ ਬਟਨ ਦੀ ਵਰਤੋਂ ਕਰਕੇ ਕੈਲਕੁਲੇਟਰ ਵਿੱਚ log35log4log35log4 ਐਕਸਪ੍ਰੈਸ਼ਨ ਦਾਖਲ ਕਰੋ। ਤਿੰਨ ਦਸ਼ਮਲਵ ਸਥਾਨਾਂ ਤੱਕ ਗੋਲ ਕਰੋ।
ਇਸਨੂੰ ਅਜ਼ਮਾਓ 10.73
ਤਿੰਨ ਦਸ਼ਮਲਵ ਸਥਾਨਾਂ ਤੱਕ ਗੋਲ ਕਰਕੇ, ਲਗਭਗ log342।
ਇਸਨੂੰ ਅਜ਼ਮਾਓ 10.74
ਤਿੰਨ ਦਸ਼ਮਲਵ ਸਥਾਨਾਂ ਤੱਕ ਗੋਲ ਕਰਕੇ, ਲਗਭਗ log546।
ਮੀਡੀਆ
ਲਘੂਗਣਕਾਂ ਦੇ ਗੁਣਾਂ ਦੀ ਵਰਤੋਂ ਕਰਨ ਵਿੱਚ ਵਾਧੂ ਨਿਰਦੇਸ਼ਾਂ ਅਤੇ ਅਭਿਆਸ ਲਈ ਇਹਨਾਂ ਔਨਲਾਈਨ ਸਰੋਤਾਂ ਤੱਕ ਪਹੁੰਚ ਕਰੋ।
ਲਘੂਗਣਕਾਂ ਨੂੰ ਵਿਸਤਾਰ ਕਰਨ ਲਈ ਲਘੂਗਣਕਾਂ ਦੇ ਗੁਣਾਂ ਦੀ ਵਰਤੋਂ ਕਰਨਾ
ਲਘੂਗਣਕਾਂ ਨੂੰ ਸੰਕੁਚਿਤ ਕਰਨ ਲਈ ਲਘੂਗਣਕਾਂ ਦੇ ਗੁਣਾਂ ਦੀ ਵਰਤੋਂ ਕਰਨਾ
ਬੇਸ ਬਦਲਣਾ
ਅਭਿਆਸ ਸੰਪੂਰਨਤਾ ਲਿਆਉਂਦਾ ਹੈ
ਲਘੂਗਣਕਾਂ ਦੇ ਗੁਣਾਂ ਦੀ ਵਰਤੋਂ ਕਰੋ
ਅਗਲੀਆਂ ਕਸਰਤਾਂ ਵਿੱਚ, ਮੁਲਾਂਕਣ ਕਰਨ ਲਈ ਲਘੂਗਣਕਾਂ ਦੇ ਗੁਣਾਂ ਦੀ ਵਰਤੋਂ ਕਰੋ।
ⓐ log41 ⓑ log88
ⓐ log121 ⓑ lne
ⓐ 3log36 ⓑ log227
1. ⓐ 5log510 ⓑ log4410
2. ⓐ 8log87 ⓑ log66−2
3. ⓐ 6log615 ⓑ log88−4
4. ⓐ 10log510 ⓑ log10−2
5. ⓐ 10log310 ⓑ log10−1
6. ⓐ eln4 ⓑ lne2
7. ⓐ eln3 ⓑ lne7
8. ਅਗਲੇ ਅਭਿਆਸਾਂ ਵਿੱਚ, ਲਘੂਗਣਕ ਦੇ ਗੁਣਨਫਲ ਗੁਣ ਦੀ ਵਰਤੋਂ ਕਰਕੇ ਹਰੇਕ ਲਘੂਗਣਕ ਨੂੰ ਲਘੂਗਣਕਾਂ ਦੇ ਜੋੜ ਵਜੋਂ ਲਿਖੋ। ਜੇ ਸੰਭਵ ਹੋਵੇ ਤਾਂ ਸਰਲ ਬਣਾਓ।
9. log 4 6 x
10. log 5 8 y
11. log 2 32 x y
12. log 3 81 x y
13. log 100 x
14. log 1000 y
15. ਅਗਲੇ ਅਭਿਆਸਾਂ ਵਿੱਚ, ਲਘੂਗਣਕ ਦੇ ਭਾਗਫਲ ਗੁਣ ਦੀ ਵਰਤੋਂ ਕਰਕੇ ਹਰੇਕ ਲਘੂਗਣਕ ਨੂੰ ਲਘੂਗਣਕਾਂ ਦੇ ਜੋੜ ਵਜੋਂ ਲਿਖੋ। ਜੇ ਸੰਭਵ ਹੋਵੇ ਤਾਂ ਸਰਲ ਬਣਾਓ।
16. log 3 3 8
17. log 6 5 6
18. log 4 16 y
19. log 5 125 x
20. log x 10
21. log 10,000 y
22. ln e 3 3
23. ln e 4 16
24. ਅਗਲੇ ਅਭਿਆਸਾਂ ਵਿੱਚ, ਲਘੂਗਣਕ ਦੇ ਘਾਤ ਗੁਣ ਦੀ ਵਰਤੋਂ ਕਰਕੇ ਹਰੇਕ ਦਾ ਵਿਸਤਾਰ ਕਰੋ। ਜੇ ਸੰਭਵ ਹੋਵੇ ਤਾਂ ਸਰਲ ਬਣਾਓ।
log 3 x 2 log 3 x 2
log 2 x 5 log 2 x 5
log x −2 log x −2
log x −3 log x −3
log 4 x log 4 x
log 5 x 3 log 5 x 3
ln x 3 ln x 3
ln x 4 3 ln x 4 3
In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.
log 5 ( 4 x 6 y 4 ) log 5 ( 4 x 6 y 4 )
log 2 ( 3 x 5 y 3 ) log 2 ( 3 x 5 y 3 )
log 3 ( 2 x 2 ) log 3 ( 2 x 2 )
log 5 ( 21 4 y 3 ) log 5 ( 21 4 y 3 )
log 3 x y 2 z 2 log 3 x y 2 z 2
log 5 4 a b 3 c 4 d 2 log 5 4 a b 3 c 4 d 2
log 4 x 16 y 4 log 4 x 16 y 4
log 3 x 2 3 27 y 4 log 3 x 2 3 27 y 4
log 2 2 x + y 2 z 2 log 2 2 x + y 2 z 2
log 3 3 x + 2 y 2 5 z 2 log 3 3 x + 2 y 2 5 z 2
log 2 5 x 3 2 y 2 z 4 4 log 2 5 x 3 2 y 2 z 4 4
log 5 3 x 2 4 y 3 z 3 log 5 3 x 2 4 y 3 z 3
In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.
log 6 4 + log 6 9 log 6 4 + log 6 9
log 4 + log 25 log 4 + log 25
log 2 80 − log 2 5 log 2 80 − log 2 5
log 3 36 − log 3 4 log 3 36 − log 3 4
log 3 4 + log 3 ( x + 1 ) log 3 4 + log 3 ( x + 1 )
log 2 5 − log 2 ( x − 1 ) log 2 5 − log 2 ( x − 1 )
log 7 3 + log 7 x − log 7 y log 7 3 + log 7 x − log 7 y
log 5 2 − log 5 x − log 5 y log 5 2 − log 5 x − log 5 y
4 log 2 x + 6 log 2 y 4 log 2 x + 6 log 2 y
6 log 3 x + 9 log 3 y 6 log 3 x + 9 log 3 y
log 3 ( x 2 − 1 ) − 2 log 3 ( x − 1 ) log 3 ( x 2 − 1 ) − 2 log 3 ( x − 1 )
log ( x 2 + 2 x + 1 ) − 2 log ( x + 1 ) log ( x 2 + 2 x + 1 ) − 2 log ( x + 1 )
4 log x − 2 log y − 3 log z 4 log x − 2 log y − 3 log z
3 ln x + 4 ln y − 2 ln z 3 ln x + 4 ln y − 2 ln z
1 3 log x − 3 log ( x + 1 ) 1 3 log x − 3 log ( x + 1 )
2 log ( 2 x + 3 ) + 1 2 log ( x + 1 ) 2 log ( 2 x + 3 ) + 1 2 log ( x + 1 )
Use the Change-of-Base Formula
In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.
log 3 42 log 3 42
log 5 46 log 5 46
log 12 87 log 12 87
log 15 93 log 15 93
log 2 17 log 2 17
log 3 21 log 3 21
Writing Exercises
Write the Product Property in your own words. Does it apply to each of the following? loga5x,loga5x,loga(5+x).loga(5+x). Why or why not?
1. ਸ਼ਕਤੀ ਗੁਣ ਨੂੰ ਆਪਣੇ ਸ਼ਬਦਾਂ ਵਿੱਚ ਲਿਖੋ। ਕੀ ਇਹ ਹੇਠ ਲਿਖਿਆਂ ਵਿੱਚੋਂ ਹਰੇਕ ਤੇ ਲਾਗੂ ਹੁੰਦਾ ਹੈ? ਲੋਗ(a)p, ਲੋਗ(a)p, (ਲੋਗ(a)x)r, (ਲੋਗ(a)x)r। ਕਿਉਂ ਜਾਂ ਕਿਉਂ ਨਹੀਂ?
ਲੋਗ(a+b) ≠ ਲੋਗ(a) + ਲੋਗ(b) ਨੂੰ ਇੱਕ ਉਦਾਹਰਨ ਨਾਲ ਦਰਸਾਓ।
1. ਆਪਣੇ ਕੈਲਕੁਲੇਟਰ ਦੀ ਵਰਤੋਂ ਕਰਕੇ ਲੌਗ715ਲੌਗ715 ਦਾ ਮੁੱਲ ਕਿਵੇਂ ਲੱਭਣਾ ਹੈ, ਇਸ ਬਾਰੇ ਦੱਸੋ।
1. ਆਤਮ-ਪੜਤਾਲ
ⓐ ਅਭਿਆਸ ਮੁਕੰਮਲ ਕਰਨ ਪਿੱਛੋਂ, ਇਸ ਭਾਗ ਦੇ ਉਦੇਸ਼ਾਂ ਉੱਤੇ ਆਪਣੀ ਮੁਹਾਰਤ ਦਾ ਮੁਲਾਂਕਣ ਕਰਨ ਲਈ ਇਸ ਚੈਕਲਿਸਟ ਦੀ ਵਰਤੋਂ ਕਰੋ।
1. ੧ ਤੋਂ ੧੦ ਦੇ ਪੈਮਾਨੇ 'ਤੇ, ਚੈਕਲਿਸਟ 'ਤੇ ਤੁਹਾਡੇ ਜਵਾਬਾਂ ਦੇ ਮੱਦੇਨਜ਼ਰ, ਤੁਸੀਂ ਇਸ ਭਾਗ ਵਿੱਚ ਆਪਣੀ ਮੁਹਾਰਤ ਨੂੰ ਕਿਵੇਂ ਦਰਜਾ ਦਿਓਗੇ? ਤੁਸੀਂ ਇਸ ਵਿੱਚ ਕਿਵੇਂ ਸੁਧਾਰ ਕਰ ਸਕਦੇ ਹੋ?