Logarithm Worksheet Express the equation in exponential form

Logarithm Worksheet Express the equation in exponential form. 2. loge 2 1/3 1, logs 25 2 Express the equation in logarithmic form. 4.8-1 1/8 3.53 z 125 Evaluate the expression. 5. (a) loge, 36 (b) log981 (c) log77 6. (a) loga (1/27) (b) logio N10 (c) logs 0.2 7. (a) 2 92 37 (b) 3 (c) e (c) 10log 87 (b) 10log5 In 8. (a) e Use the definition of the logarithmic function to find x. (b) logio o 1 x 9. (a) log15 x 4 10. (a) log4 2 x (b) log4 x 2 (b) log 25 2 11. (a) logx1000 Use a calculator to evaluate the expression, correct to four decimal places (b) In 25.3 (c) In(1 V3 12. (a) In 5 13. (a) In 27 (b) In 7.39 (c) in 54.6 Find the domain of the function. 15. f(x) log5(8- 2x) 14. f(x) logio(x 3)

Solution

1 ) We know that

The exponential form for log_b x = y is b^y= x

   log₅ 25 = 2

Exponential form is 5² = 25

2 ) log₈ 2 = 1/3

      Exponential form is  (8)^1/3 = 2

3 ) We know that

Logerithmic form for  b^y= x is   log_b x = y

5³ = 125

Logerithmic form is log₅125 = 3

4 ) 8^-1 = 1/8

   Logerithmic form is log₈1/8 = -1

5 )

a) log₆ 36 = log₆ 6² [ since log_a x^b = b log_a x ]

= 2 log₆ 6 [ since log_a a = 1 ]

= 2

Therefore,

   log₆ 36 = 2

b ) log₉ 81 = log₉ 9²   [ since log_a x^b = b log_a x ]

= 2 log₉ 9 [ since log_a a = 1 ]

= 2

Therefore,

log₉ 81 = 2

c ) log₇ 7¹⁰ = 10 log₇ 7

= 10 .1

= 10

6 )

a ) log3 (1/27) = - log₃ 27 [ since log_a (1/x) = -log_a x ]

= -log₃ 3³

= -3 log₃ 3

= -3

8 )

a ) e ^ln = e ^ln_e = [ since a^{log_a b}= b ]

   b ) 10^{log 5} = 10^{log₁₀ 5}= 5

   c ) 10^{log 87} = 10 ^{log₁₀ 87} = 87

9 )

   a ) log₅ x = 4

   5⁴ = x [ since log_b x = y   is b^y= x ]

x = 625

b ) log₁₀ 0.1 = x

10^x = 0.1

  10^x = 1/10

10^x = 10^-1 [ since if a^f(x) = a^g(x) then f(x)=g(x) ]

x = -1

10 )

a ) log₄ 2 = x

4^x = 2

(2²)^x = 2

2 ^2x = 2¹

2x = 1

x = 1/2

11 )

b) log_x 25 = 2

x² = 25

x² = 5²   

x = 5

12 )

a ) ln 5 = 1.6094

b ) ln 25.3 = 3.2308

c ) ln ( 1+ root 3 ) = 1.0051

13 ) a ) ln 27 = 3.2958

b) ln 7.39 = 2.0001

c ) ln 54.6 = 4.00003