Section 6.5 – Properties of Logarithms

𝑙𝑜𝑔𝑎1=0 𝑙𝑜𝑔𝑎𝑎=1 𝑎𝑙𝑜𝑔𝑎𝑀=M

𝑙𝑜𝑔𝑎𝑎𝑟=𝑟 𝑙𝑜𝑔𝑎 (𝑀𝑁 )=𝑙𝑜𝑔𝑎𝑀+𝑙𝑜𝑔𝑎𝑁 𝑙𝑜𝑔𝑎(𝑀𝑁 )=𝑙𝑜𝑔𝑎𝑀−𝑙𝑜𝑔𝑎𝑁

𝑙𝑜𝑔𝑎𝑀𝑟=𝑟 𝑙𝑜𝑔𝑎𝑀 𝑎𝑥=𝑒𝑥𝑙𝑛𝑎

𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔 𝒐𝒇 𝑳𝒐𝒈𝒂𝒓𝒊𝒕𝒉𝒎𝒔

𝑻𝒉𝒆𝒐𝒓𝒆𝒎𝐼𝑓 𝑀=𝑁 , h𝑡 𝑒𝑛𝑙𝑜𝑔𝑎𝑀=𝑙𝑜𝑔𝑎𝑁𝑎𝑛𝑑𝐼𝑓 𝑙𝑜𝑔𝑎𝑀=𝑙𝑜𝑔𝑎𝑁 , h𝑡 𝑒𝑛𝑀=𝑁

𝑪𝒉𝒂𝒏𝒈𝒆𝒐𝒇 𝑩𝒂𝒔𝒆𝑭𝒐𝒓𝒎𝒖𝒍𝒂𝑙𝑜𝑔𝑎𝑀=

𝑙𝑜𝑔𝑏𝑀𝑙𝑜𝑔𝑏𝑎

𝑙𝑜𝑔𝑎𝑀=ln𝑀ln𝑎

Section 6.5 – Properties of Logarithms

𝑙𝑜𝑔447 ln𝑒√2

7 𝑙𝑜𝑔44 𝑙𝑜𝑔𝑒𝑒√2

𝑒ln 23Write the following expressions as the sum or difference or both of logarithms.

√223

7

𝑙𝑜𝑔432+𝑙𝑜𝑔42𝑙𝑜𝑔4 (32∙2 )𝑙𝑜𝑔4 (64 )𝑙𝑜𝑔44

3

3 𝑙𝑜𝑔443

𝑙𝑜𝑔29 ∙ 𝑙𝑜𝑔932𝑙𝑜𝑔99𝑙𝑜𝑔92

∙ 𝑙𝑜𝑔9

32

5

1𝑙𝑜𝑔92

∙ 𝑙𝑜𝑔9

32

𝑙𝑜𝑔932𝑙𝑜𝑔92𝑙𝑜𝑔92

5

𝑙𝑜𝑔925 𝑙𝑜𝑔92𝑙𝑜𝑔92

𝑙𝑜𝑔29 ∙ 𝑙𝑜𝑔932

𝑙𝑜𝑔29 ∙𝑙𝑜𝑔232𝑙𝑜𝑔29 ❑

5

𝑙𝑜𝑔232

𝑙𝑜𝑔225

5 𝑙𝑜𝑔22

Section 6.5 – Properties of Logarithms

𝑙𝑜𝑔𝑎 (𝑥𝑦 ) 𝑙𝑜𝑔3 (𝑥 𝑦5 )𝑙𝑜𝑔𝑎 𝑥+𝑙𝑜𝑔𝑎 𝑦 𝑙𝑜𝑔3 𝑥+ 𝑙𝑜𝑔3 𝑦

5𝑙𝑜𝑔6 ( 𝑦6𝑥3 )

𝑙𝑜𝑔5(3√𝑥2+1𝑥2−1 )

Write the following expressions as the sum or difference or both of logarithms.

𝑙𝑜𝑔3 𝑥+5 𝑙𝑜𝑔3 𝑦𝑙𝑜𝑔6 𝑦

6−𝑙𝑜𝑔6 𝑥3

6 𝑙𝑜𝑔6 𝑦−3 𝑙𝑜𝑔6 𝑥

𝑙𝑜𝑔5 ( 3√𝑥2+1 )−𝑙𝑜𝑔5 (𝑥2−1 )𝑙𝑜𝑔5 (𝑥2+1 )

13−𝑙𝑜𝑔5 ( (𝑥−1 ) (𝑥+1 ) )

13𝑙𝑜𝑔

5

(𝑥2+1 )− ( 𝑙𝑜𝑔5 (𝑥−1 )+𝑙𝑜𝑔5 (𝑥+1 ) )13𝑙𝑜𝑔

5

(𝑥2+1 )−𝑙𝑜𝑔5 (𝑥−1 )− 𝑙𝑜𝑔5 (𝑥+1 )

Section 6.5 – Properties of Logarithms

5 𝑙𝑜𝑔3 𝑥+3 𝑙𝑜𝑔3 𝑦 35ln𝑥−7 ln 𝑦

𝑙𝑜𝑔3 𝑥5+ 𝑙𝑜𝑔3 𝑦

3

ln 𝑥35− 𝑙𝑛 𝑦7

Write the following expressions as a single logarithm.

ln𝑥35

𝑦7

4 𝑙𝑜𝑔9 𝑥+6 𝑙𝑜𝑔9 𝑦−3 𝑙𝑜𝑔9 𝑧

𝑙𝑜𝑔5( 𝑥2+2𝑥−3𝑥2−4 )− 𝑙𝑜𝑔5( 𝑥2+7 𝑥+6𝑥+2 )

𝑙𝑜𝑔3 (𝑥5 𝑦3 )

ln5√𝑥3𝑦7

𝑙𝑜𝑔9𝑥4+𝑙𝑜𝑔9 𝑦

6−𝑙𝑜𝑔9𝑧3

𝑙𝑜𝑔9 (𝑥4 𝑦6 )−𝑙𝑜𝑔9𝑧 3

𝑙𝑜𝑔9(𝑥4 𝑦6𝑧 3 )

𝑙𝑜𝑔5(𝑥2+2𝑥−3𝑥2−4

𝑥2+7 𝑥+6𝑥+2

)𝑙𝑜𝑔5( 𝑥2+2𝑥−3𝑥2−4

∙𝑥+2

𝑥2+7 𝑥+6 )

𝑙𝑜𝑔5( (𝑥+3 ) (𝑥−1 )(𝑥+2 ) (𝑥−2 )

∙𝑥+2

(𝑥+1 ) (𝑥+6 ) )𝑙𝑜𝑔5( (𝑥+3 ) (𝑥−1 )

(𝑥−2 ) (𝑥+1 ) (𝑥+6 ) )

Section 6.5 – Properties of Logarithms

𝑙𝑜𝑔421

Calculate the value of each expression using the Change-of-Base formula and a calculator.

ln 21ln 43.04451.38632.196

𝑙𝑜𝑔√834ln 34ln √83.52641.03973.392

𝑙𝑜𝑔 14

78

ln 78

ln14

4.3567−1.3863

−3.143

Section 6.5 – Properties of Logarithms

ln 𝑦=ln (𝑥+𝐶 )

Express each function as a function of x.

𝑦=𝑥+𝐶ln 𝑦=2 ln𝑥− ln (𝑥+1 )+ ln𝐶ln 𝑦=ln𝑥2−ln (𝑥+1 )+ln𝐶

ln 𝑦=ln 𝑥2

𝑥+1+ ln𝐶

ln 𝑦=ln𝐶 𝑥2

𝑥+1

𝑦=𝐶𝑥2

𝑥+1

ln 𝑦=5 𝑥+ln𝐶ln 𝑦=ln𝑒5 𝑥+ ln𝐶ln 𝑦=ln𝐶𝑒5𝑥

𝑦=𝐶𝑒5𝑥

Section 6.6 –Logarithmic and Exponential EquationsReview

𝑦=𝑙𝑜𝑔𝑎𝑥↔𝑥=𝑎𝑦 (𝑎>0𝑎𝑛𝑑𝑎≠1)𝐼𝑓 𝑙𝑜𝑔𝑎𝑀=𝑙𝑜𝑔𝑎𝑁 , h𝑡 𝑒𝑛𝑀=𝑁𝐼𝑓 𝑎𝑢=𝑎𝑣 , h𝑡 𝑒𝑛𝑢=𝑣

𝑙𝑜𝑔3 (𝑥−5 )=4𝑥−5=34

𝑥−5=81𝑥=86

𝑙𝑜𝑔8 (7−3𝑟 )=𝑙𝑜𝑔8 (−5𝑟 −9 )7−3𝑟=−5𝑟 −92𝑟=−16𝑟=−8

𝑙𝑜𝑔45+𝑙𝑜𝑔4 (𝑥2−1 )=𝑙𝑜𝑔415𝑙𝑜𝑔4 (5 𝑥2−5 )=𝑙𝑜𝑔4155 𝑥2−5=155 𝑥2=20𝑥2=4𝑥=±2

Examples

Section 6.6 –Logarithmic and Exponential EquationsExamples

𝑙𝑜𝑔6 (𝑥+4 )+𝑙𝑜𝑔6 (𝑥+3 )=1𝑙𝑜𝑔6 [ (𝑥+4 ) (𝑥+3 ) ]=1𝑙𝑜𝑔6 [ (𝑥+4 ) (𝑥+3 ) ]=𝑙𝑜𝑔66

(𝑥+4 ) (𝑥+3 )=6𝑥2+7 𝑥+12=6𝑥2+7 𝑥+6=0(𝑥+1 ) (𝑥+6 )=0

𝑥=−1 𝑥=−6

𝑙𝑜𝑔2 (𝑥+1 )+ 𝑙𝑜𝑔2 (𝑥+7 )=3𝑙𝑜𝑔2 [ (𝑥+1 ) (𝑥+7 ) ]=3 𝑙𝑜𝑔22

(𝑥+1 ) (𝑥+7 )=8𝑥2+8 𝑥+7=8𝑥2+8 𝑥−1=0

𝑙𝑜𝑔2 [ (𝑥+1 ) (𝑥+7 ) ]=𝑙𝑜𝑔223

𝑥=−8±√82−4 (1 ) (−1 )

2 (1 )

𝑥=−8.123 𝑥=0.123

Section 6.6 –Logarithmic and Exponential EquationsExamples

42 𝑥− 3=6442 𝑥− 3=43

2 𝑥−3=3x=3

3.82𝑥=1.7𝑥

ln 3.82𝑥=ln 1.7𝑥

2 𝑥𝑙𝑛3.8=𝑥𝑙𝑛1.72 𝑥𝑙𝑛3.8−𝑥𝑙𝑛1.7=0𝑥 (2 𝑙𝑛3.8−𝑙𝑛1.7)=0

𝑥=0

9𝑛+10+3=819𝑛+10=78𝑙𝑛9𝑛+10=𝑙𝑛78

(𝑛+10 )𝑙𝑛 9=𝑙𝑛78

(𝑛+10 )= 𝑙𝑛78𝑙𝑛9

𝑛=𝑙𝑛78𝑙𝑛 9

−10

𝑛=−8.017

Section 6.6 –Logarithmic and Exponential EquationsExamples

22 𝑥+2𝑥+2−12=0

𝑙𝑒𝑡 𝑢=2𝑥

𝑢=−6

(2𝑥)2+22 ∙2𝑥−12=0(2𝑥)2+4 ∙2𝑥−12=0

𝑢2+4𝑢−12=0(𝑢+6 ) (𝑢−2 )=0

𝑢=22𝑥=−6

ln (2¿¿𝑥 )=𝑙𝑛2¿𝑙𝑛2𝑥=ln (−6)𝑥𝑙𝑛2=𝑙𝑛2𝑥=1

2𝑥=2

Section 6.6 –Logarithmic and Exponential EquationsExample for Using a Graphing Calculator

𝑒𝑥−ln 𝑥=4

𝑥=0.053

𝑍𝑜𝑜𝑚 : 6

𝑥=1.48

𝑦 1=𝑒𝑥−ln 𝑥

𝑦 2=4

𝑍𝑜𝑜𝑚 :𝐵𝑜𝑥