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 \(\log 2 = 0.301, \log 3 = 0.477, \log_7 = 0.845\)