# 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 $$\log_{0.1}(4x^2 - 1) > \log_{0.1}3x$$