Problem
Problem
Solve the following equations.
(i) x^{1+\log_{10} x} = 10x
(ii) \log_2(9 + 2^x) = 3
(iii) 2 \cdot x^{\log_4 3} + 3^{\log_4 x} = 27
(iv) \log_4 \log_3 \log_2 x = 0
(v) \frac{\log_{10} x + 5}{x^3} = 10^{5+\log_{10} x}
(vi) \log_3 \left( \log_9 x + \frac{1}{2} + 9^x \right) = 2x
(vii) 4^{\log_{10} x+1} - 6^{\log_{10} x} - 2 \cdot 3^{\log_{10} x^2+2} = 0
(viii) \frac{\log_{10}(x-3)}{\log_{10}(x^2-21)} = \frac{1}{2}
(ix) x^{\log_2 x+4} = 32
(x) \log_a x = x, \text{ where } a = x^{\log_4 x}
(xi) \log_{\sqrt{2}} \sin x (1 + \cos x) = 2
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