Logarithms

 

2.Solve for n:  3ⁿ = 5ⁿ⁻²  [ AP SSC ]
3.If x= log 23,  y=log 2 5 then express  log 2 15 in terms of x and y
4.Show that log 1000 = 3 log 2 + 3 log 5
5.Write 2 log 3 + 3 log 5 – 5 log 2 as a single logarithm
6.If log[(x+y)/3]  = ½log x + log y then prove that   (x/y) + (y/x)  = 7
7.If  log 2√3 1728 = x, then find the value of  x.
8. Expand log(343/125)  [AP SSC 2016M] 
9. Find the value of  log 81 3   [AP SSC 2016M]

 

Objective Questions

 Exponential Form to Log form 

1.The logarithmic form of  8x = 2   is  

  1. log x 8 =2  
  2. log  = x
  3. log x 2 =8 
  4. log 2 x = 8

1.The value of log 10 10 =

2.Log 10, log 100, log 1000……. are in  …………………….. progression.

 Log form to  Exponential Form

1.The exponential form of log48 = x  is    [AP SSC 2016 M]

  1.   x8 = 4
  2.  x4 =8
  3.  4x =8
  4.   8x =4

 2.The exponential form of log10 0.001= -3 is .... [TS SSC 2015M]

  1. (0.001)¹⁰ =-3        
  2.  (-3)¹⁰ =0.001    
  3. 10³  =  -0.001  
  4.  10⁻³   =0.001

3. If log√3 x = 4  then  the value of x is   

  1.   9
  2. 10
  3. 11
  4. 13

4. log2 (sin 90º ) = 1 Is this   true or false ? [TS-SSC-2016J]

5.The value of   log 255 is    [AP SSC 2017 M]

  1. 1/2  
  2. 2 
  3. 5 
  4. 25
6.Find the value of   log 6 36   [SCERT MP 2022 ]
7.The value  of log100.01    is ______ [ AP SCERT MP ]
8. The value of log10001    is ______     [ AP SCERT MP ]
9.The logarithmic form of  210 = 1024   is ______    [SCERT MP ]
10.Determine the value of  log 2512  
11.Find the value of   log 5√625  [ AP SSC 2019 M]

12.Determine the value of  log 1 1    

  1. 0
  2. 1
  3. 0 or 1 
  4. undefined

Properties of Logarithms

1.The expansion of  log 10000 is _____ [AP SCERT MP ] 

  1.  a4(log 5+  log 2 )  
  2.   log 125 + log 800

2.The short form of log16 - 2 log 2  is ____   [ AP SCERT MP] 

  1. log 4   
  2. log12  
  3. log 32  
  4. log8

3.The value of   log 10 2 +log 105 is ...........  [TS SSC 2017 J]

  1. 1 
  2. 2
  3. 5 
  4. 10

4.Which of the following is not a rational number? 

  1. log 10 2 
  2. 1.020202020202020202...........
  3. 102.102 
  4. 10/19

5.which of the following is not a rational number? [TS-2016M]

  1. log₁₀³ 
  2. 5.23232323....
  3. 123.123 
  4. 10/19

6.Which of the following is a rational number?   

  1. π 
  2. e 
  3. 22/7     
  4. log 32  

7.Find the value of x which satisfies the relation log₁₀3+log₁₀(4x+1)=log₁₀(x+1) +1 [MBA 2004]

8.If log₂ [log₃ (log₂ x)]=1, then find the value of x   [MBA 2005]

9. If log₁₀ (x²-6x+45) =2, find the value of x [ MBA 2006 ] 

10. What is the value of  [ log₁₀ (5log₁₀ 100)]²   [MBA 2010 ] 

  1. 1
  2. 2
  3. 10
  4. 25

11.If 3log(x+3) = log 27, then the value of x is.....  [TS SSC 2016J]

  1. 24   
  2.  6   
  3. 3
  4.  0

12. If  logₘn  =½ , logₙp = ⅓  and logₚm =k/5, find the value of k  [ MAT 2006] 

13If x²+y² = 10xy, then prove that, 2log(x+y) = log x + log y + 2log2 + log 3  [TS SSC 2023 A]

Other Competitive Exams 

1.If  log₄log₂log₃(x–2009) =0 then the value of ‘x’ is   [TET cum TRT 2018]

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