Logarithms Practice Problems

Here you will find a concise collection of logarithms practice problems. Visit this page directly at hunkim.com/logarithms or hunkim.com/log

1. Describe the connection between 2^3=8 and \log_2 8=3
   Solution
   In general a^b=c is equivalent to \log_a c=b
2. Write as a logarithm: 7^{-2}=\frac{1}{49}
   Solution
   \log_7 \frac{1}{49}=-2
3. Evaluate:
   1. \log 100
      Solution
      2
   2. \log 1,000,000
      Solution
      6
   3. \log 10
      Solution
      1
   4. \log 1
      Solution
      0
   5. \log{\frac{1}{1000}}
      Solution
      -3
   6. \log_3 81
      Solution
      4
      
      
4. Evaluate 3\log_2 1024
   Solution
   30
   
   
5. y=\log x
   1. Sketch and label 3 points
      Solution
      
      
   2. What is the assumed base?
      Solution
      10
      
      
   3. How does this graph compare with y=\log_2 x or y=\log_5 x?
      Solution
      Same x-intercept but the graph curves more sharply
      
      
   4. Intercept(s)?
      Solution
      x=0
      
      
6. Enrichment – the natural logarithm:
   1. Define y=\ln x
      Solution
      y=\log_e x
      
      
   2. Evaluate \ln e
      Solution
      1
      
      
   3. Evaluate \ln\left(\frac{1}{e^3}\right)
      Solution
      -3
      
      
7. What are the restrictions on y=\log_b x?
   Solution
   b, x>0. Also, b\neq 1 except for the case \log_1 1=1
   
   
8. Basic log rules:
   1. \log x^3=k \log x. Find k
      Solution
      3
      
      
   2. \log_2 x=\frac{\log x}{k}. Find k
      Solution
      \log 2
      
      
   3. \log_2 3=\frac{k}{\log 2}. Find k
      Solution
      \log 3
      
      
   4. \log_{7} 5=\frac{\log_{3} 5}{k}. Find k
      Solution
      \log_3 7
      
      
   5. \log_2(15)=\log_2 3+k. Find k
      Solution
      \log_2 5
      
      
   6. \log_7\left(\frac{2}{5}\right)=\log_7 2-k. Find k
      Solution
      \log_7 5
      
      
9. Enrichment – Prove the following log rules:
   1. Product property \log_b(xy)=\log_b x+\log_b y
   2. Quotient property \log_b\left(\frac{x}{y}\right)=\log_b x-\log_b y
   3. Power property \log_b \left(x^k\right)=k\log_b x
   4. Change of base property \log_a x=\frac{\log_b x}{\log_b a}
      
10. Basic log transformations:
    1. y=\log x+2
       Solution
       Shift 2 units up
    2. y=\log(x+3)
       Solution
       Shift 3 units left
    3. y=2\log(x-1)+5
       Solution
       Multiply y’s by 2, then shift 1 right and 5 up
    4. y=-\log(2x)
       Solution
       Multiply y’s by -1 and multiply x’s by \frac{1}{2}
    5. Challenge – sketch f(x)=\sqrt{\ln (x-2)}
       Solution
       
       
       
       
11. What is the equation of the log function below?
    
    Solution
    y=\log\left(\frac{1}{2}(x-2)\right)
    
    
12. f(x)=-8\log_5 [10(x+8)]
    1. Range?
       Solution
       All real numbers
       
       
    2. Domain?
       Solution
       x>-8
       
       
    3. Equation of the asymptotes?
       Solution
       x=-8
       
       
    4. x-intercept?
       Solution
       -7.9
       
       
13. Write as a single logarithm: \log_4 a+7\log_4 b+\log_4 z
    Solution
    \log_4 ab^7z
    
    
14. If a=\log 3, b=\log 5, c=\log 7, rewrite \log \frac{5}{441} in terms of a, b, and c. Hint: 21\times21=441
    Solution
    -2a+b-2c
    
    
15. Evaluate: \log_3 27-\log_x 1+\log_7 7
    Solution
    4
    
    
16. \log_{\sqrt{2}}\left(\frac{1}{x}\right)=k\log_2 x. Find k
    Solution
    -2
    
    
17. Given \log_7 21\approx 1.56 estimate \log_7 3 without a calculator
    Solution
    0.56
    
    
18. Find the inverse of:
    1. f(x)=\log x?
       Solution
       10^x
       
       
    2. y=\log_2 (x-3)?
       Solution
       2^x+3
       
       
    3. y=2^x?
       Solution
       \log_2 x
       
       
    4. y=e^x?
       Solution
       \ln x
       
       
    5. f(x)=\left (\frac{1}{6} \right)^x
       Solution
       -\log_6 x or \log_6\left(\frac{1}{x}\right)
       
       
19. True or False:
    1. y=\log x^3=3\log x
       Solution
       True
       
       
    2. y=\log x^2=2\log x
       Solution
       False. Only the right half is equivalent
       
       
    3. y=\log (x^2)=\log (x)^2
       Solution
       False
       
       
    4. \log_2 x cannot be negative
       Solution
       False
       
       
    5. y=\log (-x) is undefined
       Solution
       False. We mirror f(x)=\log(x) about the y-axis
       
       
    6. \log_2 5+\log x=\log_2 5x
       Solution
       False
       
       
    7. \log(xy^2)=2\log(xy)
       Solution
       False
       
       
    8. f^{-1}(x)=\frac{1}{f(x)}
       Solution
       False
       
       
20. Simplify 3^{\log_3 9}
    Solution
    9
    
    
21. Given 5^{2x-1}=3^{x+2}Show that x=\frac{-2\log 3-\log 5}{\log 3-2\log 5}
    1. Show that x=\frac{\log 5+2\log 3}{2\log 5-\log 3}
    2. Find k given x=\log_k 45
    3. Solution
       \frac{25}{3}
       
       
22. Solve the log equation:
    1. \log_3 x=\log_3 2+\log_3 3
       Solution
       6
       
       
    2. \log(x+2)+\log(x-1)+1=2
       Solution
       3
       
       
    3. 3+3\log(2x)=15
       Solution
       5000
       
       
    4. \log(x+2)+3+\log(x-1)=4
       Solution
       3. Reject the extraneous root
       
       
    5. \log_2(x-6)=3-\log_2(x-4)
       Solution
       8. Reject the extraneous root
       
       
    6. \log_2(x+3)^2=4
       Solution
       1 or -7. Do not drop the exponent first in this question (or you will lose the second root)
       
       
    7. Challenge: e^{\ln x^2}=\pi^2
       Solution
       \pm\pi
       
       
23. Applications of Logarithms:
    1. How many times stronger is an earthquake that has a magnitude of 8.5 vs 6.5? Hint: M=\log\left(\frac{I_A}{I_0}\right)
       Solution
       Magnitude 8.5 is 100 times stronger than magnitude 6.5
       
       
24. Challenge – solve \log\left(\frac{9}{10}\right)<\frac{9}{10}
    Solution
    x>\frac{9}{10\log\left(\frac{9}{10}\right)} or with a non-graphing calculator x>-19.7
    
    

• Click here for more IB Math SL style logarithms practice problems