Indices and Logarithms Quiz - ICSE Class 10 Mathematics

Free practice quiz: Indices and Logarithms (ICSE Class 10 Mathematics). 47 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 10 Mathematics practice quiz has 47 multiple-choice questions on Indices and Logarithms, and 47 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. If $a^x = b$, $b^y = c$, and $c^z = a$, then find the value of $xyz$.

   1. 1
   2. 0
   3. -1
   4. 2
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   Answer: A. 1

   Given $a^x = b$. Substituting this into $b^y = c$, we get $(a^x)^y = c$, which simplifies to $a^{xy} = c$. Substituting this into $c^z = a$, we get $(a^{xy})^z = a$, which simplifies to $a^{xyz} = a^1$. Therefore, $xyz = 1$.

2. If $9^{x+1} = 27^{x-2}$, find $x$.

   1. 7
   2. 8
   3. 9
   4. 10
   Show answer

   Answer: B. 8

   We can rewrite both 9 and 27 as powers of 3. $9 = 3^2$ and $27 = 3^3$. So the equation becomes $(3^2)^{x+1} = (3^3)^{x-2}$. This simplifies to $3^{2(x+1)} = 3^{3(x-2)}$. Equating the exponents: $2(x+1) = 3(x-2)$. $2x+2 = 3x-6$. Subtracting $2x$ from both sides gives $2 = x-6$. Adding 6 to both sides gives $x = 8$.

3. If $\log_a 25 = 2$, what is $a$?

   1. 3
   2. 4
   3. 5
   4. 6
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   Answer: C. 5

   The equation $\log_a 25 = 2$ means $a^2 = 25$. Taking the square root of both sides, we get $a = \pm 5$. However, the base of a logarithm must be positive and not equal to 1. Therefore, we choose the positive value, $a=5$.

4. Evaluate: $\log_3 27 - \log_3 3$.

   1. 1
   2. 2
   3. 3
   4. 4
   Show answer

   Answer: B. 2

   We know that $27 = 3^3$, so $\log_3 27 = 3$. Also, $\log_3 3 = 1$. Therefore, $\log_3 27 - \log_3 3 = 3 - 1 = 2$. Alternatively, using the property $\log_b m - \log_b n = \log_b (m/n)$, we have $\log_3 27 - \log_3 3 = \log_3 (27/3) = \log_3 9$. Since $9 = 3^2$, $\log_3 9 = 2$.

5. If $100^x = 1000$, find $x$.

   1. 1/2
   2. 3/2
   3. 2/3
   4. 3
   Show answer

   Answer: B. 3/2

   We can rewrite both 100 and 1000 as powers of 10. $100 = 10^2$ and $1000 = 10^3$. So the equation becomes $(10^2)^x = 10^3$. This simplifies to $10^{2x} = 10^3$. Equating the exponents: $2x = 3$. Dividing by 2: $x = 3/2$.

More questions in this quiz

Answers and explanations are in the full quiz.

1. If $3^{x+1} = 9^{x-2}$, then find the value of $x$.
2. If $\log_x y = 2$ and $\log_y x = 1/2$, what is the relationship between $x$ and $y$?
3. What is the value of $\log_a (a^2) + \log_a (a^3)$?
4. If $10^x = 0.0001$, what is the value of $x$?
5. If $\log_a b = 2$ and $\log_b c = 3$, find $\log_a c$.
6. If $\log_3 (x) = 2$, what is $x$?
7. If $2^x = 16$, what is $x$?
8. Solve for $y$: $\log_5 (y-2) = 1$.
9. What is the value of $\log_{10} 1000$?
10. If $\log_2 (x+3) = 3$, find $x$.
11. Simplify: $\frac{a^5 b^{-2} c^3}{a^2 b^3 c^{-1}}$
12. If $\log_b x = 0.5$, what is $x$ in terms of $b$?
13. Evaluate: $8^{\log_8 5}$.
14. What is the value of $\log_b (b^n)$?
15. If $\log x = 2$, what is $x$?
16. Solve for $z$: $3^{\log_3 z} = 7$.
17. If $10^{\log_{10} 5} = x$, what is $x$?
18. Solve for $x$: $2^{x+3} = 4^{x-1}$.
19. What is the value of $\log_a 1$?
20. If $\log_3 x = 4$, what is $x$?
21. If $\log_x y = 3$, then $y$ in terms of $x$ is:
22. What is the value of $\log_7 1$?
23. If $2^x = 32$, find $x$.
24. Solve for $x$: $\log (2x-1) = 2$. (Assume base 10)
25. If $\log_a 16 = 2$, what is $a$?
26. If $\log_2 (x-1) = 3$, find $x$.
27. What is the value of $\log_b b$?
28. If $3^{\log_3 8} = x$, what is $x$?
29. Solve for $y$: $\log_4 (y+2) = 2$.
30. If $\log_a b = 2$ and $\log_b c = 3$, find $\log_c a$.
31. Simplify: $5^x \cdot 5^y$.
32. If $\log_2 32 = x$, find $x$.
33. Solve for $x$: $\log_5 x = 2$.
34. Evaluate: $\log_4 8$.
35. If $\log_3 5 = a$ and $\log_3 2 = b$, find $\log_3 10$.
36. If $a^x = b$, $b^y = c$, and $c^z = a^5$, then find the value of $xyz$.
37. Solve for $x$: $\log_2 (x+4) = 3$.
38. What is the value of $\log_a a^5$?
39. If $10^x = 10000$, find $x$.
40. Solve for $x$: $\log_3 (2x+1) = 2$.
41. If $\log x + \log 5 = \log 50$, find $x$.
42. If $a^x = b$, $b^y = c$, and $c^z = a^k$, find $k$ if $xyz = 10$.

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