# 10. Logarithm Problems Part 1

1. Find the value of $$x$$ where $$\log_{\sqrt{8}} x = \frac{10}{3}$$

2. Prove that $$\log_b a.\log_c b.\log_a c = 1$$

3. Prove that $$\log_3 \log_2 \log_{\sqrt{5}} (625) = 1$$

4. If $$a^2 + b^2 = 23ab,$$ then prove that $$\log \frac{a + b}{5} = \frac{1}{2}(\log a + \log b)$$

5. Prove tha $$7\log \frac{16}{15} + 5\log \frac{25}{24} + 3\log \frac{81}{80} = \log 2$$

6. Find the value of $$\log\tan 1^{\circ} + \log\tan 2^{\circ} + \ldots + \log \tan 89^{\circ}$$

7. Evaluate $$\log_9 \tan \frac{\pi}{6}$$

8. Evaluate $$\frac{\log_{a^2}b}{\log_{\sqrt{a}}(b)^2}$$

9. Evaluate $$\log_{\sqrt{5}}.008$$

10. Evaluate $$\log_{2\sqrt{3}}144$$

11. Prove that $$\log_3 \log_2 \log_{\sqrt{3}} 81 = 1$$

12. Prove that $$\log_a x \log_b y = \log_b x \log_a y$$

13. Prove that $$\log_2 \log_2 \log_2 16 = 1$$

14. Prove that $$\log_a x = \log_b x\log_c b \ldots \log_n m \log_a n$$

15. Prove that $$a^x = 10^x\log_{10}a$$

16. If $$a^2 + b^2 = 7ab,$$ prove that $$\log \left\{\frac{1}{3}(a + b)\right\} = \frac{1}{2}(\log a + \log b)$$

17. Prove that $$\frac{\log a(\log_b a)}{\log b(\log_a b)} = -\log_a b$$

18. Prove that $$\log(1 + 2 + 3) = \log 1 + \log 2 + \log 3$$

19. Prove that $$2\log(1 + 2 + 4 + 7 + 14) = \log 1 + \log 2 + \log 4 + \log 7 + \log 14$$

20. Prove that $$\log 2 + 16\log \frac{16}{15} + 12\log \frac{25}{24} + 7\log \frac{81}{80} = 1$$

21. Simplify $$\frac{\log_9 11}{\log_5 13} \div \frac{\log_3 11}{\log_{\sqrt{5}} 13}$$

22. Simplify $$3^{\sqrt{\log_3 2}} - 2^{\sqrt{\log_2 3}}$$

23. Find the least integer $$n$$ such that $$7^n > 10^5,$$ given that $$\log_{10} 343 = 2.5353$$

24. if $$a, b, c$$ are in G.P., prove that $$\log_a x, \log_b x, \log_c x$$ are in H.P.

25. Prove that $$\log \sin 8x = 3\log 2 + \log\sin x + \log\cos x + \log\cos 2x + \log\cos 4x$$

26. If $$x = \log_{2a} a, y = \log_{3a}2a$$ and $$z = \log_{4a} 3a,$$ then prove that $$xyz + 1 = 2yz$$

27. If $$a$$ and $$b$$ are the lengths of the sides and $$c$$ be the length of the hypotenuse of a right triangle and $$c - b \neq 1$$ and $$c + b neq 1,$$ prove that $$\log_{c + b} a + \log_{c - b} a = 2\log_{c + b}a\log_{c -b}a.$$

28. If $$\frac{\log x}{y - z} = \frac{\log y}{z - x} = \frac{\log z}{x - y},$$ then prove that $$x^xy^yz^z = 1$$

29. If $$\frac{yz\log(yz)}{y + z} = \frac{zx\log(zx)}{z + x} = \frac{xy\log(xy)}{x + y},$$ prove that $$x^x = y^y = z^z$$

30. Prove that $$(yz)^{\log y - \log z}(zx)^{\log z - \log x}(xy)^{\log x -\log y} = 1$$

31. Prove that $$\frac{1}{\log_2 N} + \frac{1}{\log_3 N} + \ldots + \frac{1}{\log_{1988} N} = \frac{1}{\log_{1988!} N}$$

32. If $$0 < x <1,$$ prove that $$\log(1 + x) + \log(1 + x^2) + \log(1 + x^4) \ldots$$ to $$\infty = -\log(1 - x)$$

33. Find the sum of the series $$\frac{1}{\log_2 a} + \frac{1}{\log_4 a} + \ldots$$ upto $$n$$ terms.

34. If $$\log_4 10 = x, \log_2 20 = y$$ and $$\log_5 8 = z,$$ prove that $$\frac{1}{x + 1} + \frac{1}{y + 1} + \frac{1}{z + 1} = 1.$$

35. If $$x = \log_a(bc), y = \log_b(ca), z = \log_c(ab),$$ prove that $$\frac{1}{x + 1} + \frac{1}{y + 1} + \frac{1}{z + 1} = 1.$$

36. Prove that $$\frac{1}{1 + \log_b a + \log_b c} + \frac{1}{1 + \log_c a + \log_c b} + \frac{1}{1 + \log_a b + \log_a c} = 1.$$

37. Prove that $$x^{\log y - \log z}y^{\log z - \log x}z^{\log x - \log y} = 1$$

38. If $$\frac{\log a}{y - z} = \frac{\log b}{z - x} = \frac{\log c}{x - y},$$ prove that $$a^xb^yc^z = 1$$

39. If $$\frac{x(y + z - x)}{\log x} = \frac{y(z + x - y)}{\log y} = \frac{z(x + y - z)}{\log z},$$ prove that $$y^zz^y = z^xx^z = x^yy^x$$

40. If $$\frac{\log a}{b - c} = \frac{\log b}{c - a} = \frac{\log c}{a - b},$$ prove that $$a^{b + c}b^{c + a}c^{a + b} = 1.$$

41. If $$\frac{\log x}{q - r} = \frac{\log y}{r - p} = \frac{\log z}{p - q},$$ prove that $$x^{q + r}y^{p + r}z^{p + q} = x^py^qz^r$$

42. If $$y = a^{\frac{1}{1 - \log_a x}}$$ and $$z = a^{\frac{1}{1 - \log_a y}},$$ prove that $$x = a^{\frac{1}{1 - \log_a z}}$$

43. Let $$f(x) = \frac{1}{1 - \log_e x}.$$ If $$f(y) = e^{f(z)}$$ and $$z= e^{f(x)},$$ prove that $$x = e^{f(y)}$$

44. Show that $$\frac{1}{\log_2 n} + \frac{1}{\log_3 n} + \frac{1}{\log_4 n} + \ldots + \frac{1}{\log_{43} n} = \frac{1}{\log_{43!}n}$$

45. Show that $$2(\log a + \log a^2 + \log a^3 + \ldots + \log a^n) = n(n + 1)\log a$$

46. Find the number of digits in $$12^{12},$$ without actual computation. [ Given $$\log 2 = .301$$ and $$\log 3 = .477$$ ]

47. How many positive intergers have characteristics $$2$$ when base is $$3.$$

48. Prove that $$\log_a x \log_b y = \log_b x \log a y$$

49. If $$a, b, c$$ are in G.P., prove that $$\log_a x, \log_b y, \log_c z$$ are in H.P.

50. How many zeroes are there between the decimal point and the first sinificant digit in $$0.0504^{10}.$$ Given $$\log 2 = 0.301, \log 3 = 0.477, \log_7 = 0.845$$