14. Logarithm Problems Part 3

1. Solve $$\log_2(x^2 - 24) > \log_2 5x$$

2. Show that $$\frac{1}{\log_3\pi} + \frac{1}{\log_4\pi} > 2$$

3. Without actual computation find greater among $$(0.01)^{\frac{1}{3}}$$ and $$(0.001)^{\frac{1}{5}}$$

4. Without actual computation find greater among $$\log_2 3$$ and $$\log_3 11$$

5. Solve for $$x, \log_3(x^2 + 10) > \log_3 7x$$

6. Solve $$x^{\log_{10} x} > 10$$

7. Solve $$\log_2 x\log_{2x} 2\log_2 4x > 1$$

8. Solve $$\log_2 x\log_3 2x + \log_3 x\log_2 4x > 0$$

9. Find the value of $$\log_{12}60$$ if $$\log_6 30= a$$ and $$\log_{15}24 = b$$

10. If $$\log_ax, \log_bx$$ and $$\log_cx$$ be in A.P. and $$x\neq 1,$$ prove that $$c^2 = (ac)^{\log_a b}$$

11. If $$a = \log_{\frac{1}{2}}(\sqrt{0.125})$$ and $$b = \log_3\left(\frac{1}{\sqrt{24} - \sqrt{17}}\right)$$ then find whether $$a > 0, b > 0$$ or not.

12. Which one if greater among $$\cos(\log_e\theta)$$ or $$\log_e(\cos\theta)$$ if $$e^{\frac{-\pi}{2}} < \theta < \frac{\pi}{2}$$

13. If $$\log_2 x + \log_2 y \geq 6,$$ prove that $$x + y \geq 16$$

14. If $$a, b, c$$ be three distinct positive numbers, each different from $$1$$ such that $$\log_b a \log_c a - \log_a a + \log_a b\log_c b - \log_b b + \log_a c \log _b c - \log_c c = 0$$

15. If $$y = 10^{\frac{1}{1 - \log x}}$$ and $$z = 10^{\frac{1}{1 - \log y}},$$ prove that $$x = 10^{\frac{1}{1 - \log z}}$$

16. If $$n$$ is a natural number such that $$n = p_1^{a_1}p_2^{a_2}p_3^{a_3} \ldots p_k^{a_k}$$ and $$p_1, p_2, p_3, \ldots, p_k$$ are distinct primes, then show that $$\log n \geq k \log 2$$

17. The numbers $$3, 3\log_y x, 3\log_z y, 7\log_x z$$ form an A.P. Prove that $$x^{18} = y^{21} = z^{28}$$

18. Prove that $$\log_4 18$$ is an irrational number.

19. If $$x, y, z > 1$$ are in G.P. then prove that $$\frac{1}{1 + ln x}, \frac{1}{1 + ln y}, \frac{1}{1 + ln z}$$ are in H.P.

20. Find the value of $$\log_{30} 8,$$ if $$\log_{30}3 = a$$ and $$\log_{30}5 = b$$

21. Find the value of $$\log_{54}168,$$ if $$\log_7 12 = a$$ and $$\log_{12} 24 = b$$

22. If $$a\neq 0$$ and $$\log_x (a^2 + 1) < 0$$ then find the interval in which $$x$$ lies.

23. If $$\log_{12}18 = a$$ and $$\log_{24}54 = b,$$ prove that $$ab + 5(a - b) = 1$$

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

25. If $$a, a_1, a_2, \ldots, a_n$$ are in G.P. and $$b, b_1, b_2, \ldots, b_n$$ in A.P. with positive terms and also the common difference of A.P. and common ratios of G.P. are positive, show that there exists a system of logarithm for which $$\log a_n - b_n = = \log a - b$$ for any $$n$$. Find base $$b$$ of the system.

26. If $$\log_3 2, \log_3(2^x - 5)$$ and $$\log_3\left(2^x - \frac{7}{2}\right)$$ are in A.P., find the value of $$x.$$

27. Prove that $$\log_2 7$$ is an irrational number.

28. If $$\log_{0.5}(x - 2) < \log_{0.25}(x - 2),$$ then find the interval in which $$x$$ lies.