14. Logarithm Problems Part 3

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

102. Show that \(\frac{1}{\log_3\pi} + \frac{1}{\log_4\pi} > 2\)

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

104. Without actual computation find greater among \(\log_2 3\) and \(\log_3 11\)

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

106. Solve \(x^{\log_{10} x} > 10\)

107. Solve \(\log_2 x\log_{2x} 2\log_2 4x > 1\)

108. Solve \(\log_2 x\log_3 2x + \log_3 x\log_2 4x > 0\)

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

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

111. 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.

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

113. If \(\log_2 x + \log_2 y \geq 6,\) prove that \(x + y \geq 16\)

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

115. If \(y = 10^{\frac{1}{1 - \log x}}\) and \(z = 10^{\frac{1}{1 - \log y}},\) prove that \(x = 10^{\frac{1}{1 - \log z}}\)

116. 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 \(\log n \geq k \log 2\)

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

118. Prove that \(\log_4 18\) is an irrational number.

119. 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.

120. Find the value of \(\log_{30} 8,\) if \(\log_{30}3 = a\) and \(\log_{30}5 = b\)

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

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

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

124. If \(a, b, c\) are in G.P. show that \(\log_a x, \log_b x, \log_c x\) are in H.P.

125. 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.

126. 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 \(x.\)

127. Prove that \(\log_2 7\) is an irrational number.

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