12. Logarithm Problems Part 2

  1. Find the number of digits in \(72^{15}\) without actual computation. Given \(\log 2 = 0.301, \log 3 = 0.477\)

  2. How many positive integers have characteristics \(2\) when base is \(5\)?

  3. If \(\log 2 = 0.301\) and \(\log 3 = 0.477,\) find the number of digits in \(3^{15}\times 2^{10}\)

  4. If \(\log 2 = 0.301\) and \(\log 3 = 0.477,\) find the number of digits in \(6^{20}\)

  5. If \(\log 2 = 0.301\) and \(\log 3 = 0.477,\) find the number of digits in \(5^{25}\)

  6. Solve \(\log_a [1 + \log_b \{1 + \log_c (1 + \log_p x)\}] = 0\)

  7. Solve \(\log_7\log_5(\sqrt{x + 5} + \sqrt{x}) = 0\)

Solve following equations:

  1. \(\log_2 x + \log_4 (x + 2) = 2\)

  2. \(\log_(x + 2)x + \log_x (x + 2) = \frac{5}{2}\)

  3. \(\frac{\log (x + 1)}{\log x} = 2\)

  4. \(2\log_x a + \log_{ax} a + 3\log_{a^2x} a = 0 [a > 0]\)

  5. \(x + \log_{10}(1 + 2^x) = x\log_{10}5 + \log_{10}6\)

  6. \(x^{\frac{3}{4}(\log_2 x)^2 + \log_2x - \frac{5}{4}} = \sqrt{2}\)

  7. \((x^2 + 6)^{\log_3 x} = (5x)^{\log_3 x}\)

  8. \((3 + 2\sqrt{2})^{x^2 - 6x + 9} + (3 - 2\sqrt{2})^{x^2 - 6x + 9} = 6\)

  9. \(\log_8\left(\frac{8}{x^2}\right) \div (\log_8 x)^2 = 3\)

  10. \(\sqrt{\log_2 (x)^4} + 4\log_4\sqrt{\frac{2}{x}} = 2\)

  11. \(2\log_{10}x - \log_x0.01 = 5\)

  12. \(\log_{\sin x}2\log_{\cos x}2 + \log_{\sin x} 2 + \log_{\cos x}2 = 0\)

  13. Solve \(2^{x + 3} + 2^{x + 2} + 2^{x + 1} = 7^x + 7^{x - 1}\)

  14. \(\log_{\sqrt{2}\sin x}(1 + \cos x) = 2\)

  15. \(\log_{10}[198 + \sqrt{x^3 - x^2 - 12x + 36}] = 2\)

  16. If \(\log 2 = .30103\) and \(\log 3 = .47712,\) solve the equation \(2^x3^{2x} - 100 = 0\)

  17. Solve \(\log_x 3\log_{\frac{x}{3}}3 + \log{\frac{x}{81}}3 = 0\)

  18. Solve for \(x\) the following equation:

    \(\log_{(2x + 3)}(6x^2 + 23x + 21) = 4 - \log_{(3x + 7)}(4x^2 + 12x + 9)\)

  19. Solve the equation \(\log_2(x^2 - 1) = \log_{\frac{1}{2}}(x - 1)\)

  20. Solve \(\log_5\left(5^{\frac{1}{x} + 125}\right) = \log_5 6 + 1 + \frac{1}{2x}\)

  21. Solve the following equation for:math:x and \(y\)

    \(\log_{100}|x + y| = \frac{1}{2}\) and \(\log_{10} y - \log_{10}|x| = \log_{100} 4\)

  22. Solve \(2\log_2\log_2 x + \log_{\frac{1}{2}}\log_2(2\sqrt{2}x) = 1\)

  23. Solve \(\log_{\frac{3}{4}}\log_8(x^2 + 7) + \log_{\frac{1}{2}}\log_{\frac{1}{4}}(x^2 + 7)^{-1} = -2\)

  24. Solve for \(x\) and \(y\)

    \(\log_{10}x + \log_{10}x^{\frac{1}{2}} + \log_{10}x^{\frac{1}{4}} \ldots\) to \(\infty = y\)

    \(\frac{1 + 3 + 5 + \ldots + (2y - 1)}{4 + 7 + 10 + \ldots + (3y + 1)} = \frac{20}{7\log_10 x}\)

  25. Solve \(18^{4x - 3} = (54\sqrt{2})^{3x - 4}\)

  26. Solve \(4^{\log_9 3} + 9^{\log_2 4} = 10^{\log_x 83}\)

  27. Solve \(3^{4\log_9 (x + 1)} = 2^{2\log_2 (x + 3)}\)

  28. Solve \(\frac{6}{5}a^{\log_a x\log_{10} a \log_a 5} - 3^{\log_{10}\left(\frac{x}{10}\right)} = 9^{\log_{100}x + \log_4 2}\)

  29. Solve \(2^{3x + \frac{1}{2}} + 2^{x + \frac{1}{2}} = 2^{\log_2 6}\)

  30. Solve \((5 + 2\sqrt{6})^{x^2 - 3} + (5 - 2\sqrt{6})^{x^2 - 3} = 10\)

  31. For \(x > 1,\) show that \(2\log_{10}x - \log_x .01 \geq 4\)

  32. Show that \(|\log_b a + \log_a b| > 2\)

  33. Solve \(\log_{0.3}(x ^2 + 8) > \log_{0.3}9x\)

  34. Solve \(\log_{x - 2}(2x - 3) > \log(x - 2)(24 - 6x)\)

  35. Find the interval in which \(x\) will lie if \(\log_{0.3}(x - 1)< \log_{0.09}(x - 1)\)

  36. Solve \(\log_{\frac{1}{2}}x \geq \log_{\frac{1}{3}}x\)

  37. Solve \(\log_{\frac{1}{3}}\log_4(x^2 - 5) > 0\)

  38. Solve \(\log (x^2 - 2x - 2)\leq 0\)

  39. Solve \(\log_2^2(x - 1)^2 - \log_{0.5}(x - 1) > 5\)

  40. Prove that \(\log_2 17\log{\frac{1}{5}} 2\log_3\frac{1}{5} > 2\)

  41. Show that \(\log_{20} 3\) lies between \(\frac{1}{2}\) and \(\frac{1}{3}\)

  42. Show that \(\log_{10}2\) lies between \(\frac{1}{4}\) and \(\frac{1}{3}\)

  43. Solve \(\log_{0.1}(4x^2 - 1) > \log_{0.1}3x\)