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Logarithmic Functions

Logarithmic Functions - Solving Equations


If xx satisfies the equation logx=log322log492\log{x} = \log{322} - \frac{\log{49}}{2}, what is the value of xx?

Suppose that 0xπ0 \leq x \leq \pi satisfies logsinxcosx+logcosxtanx=1. \log_{\sin x}{\cos x} + \log_{\cos x}{\tan x} = 1. If x=abπx = \frac{a}{b}\pi, where aa and bb are coprime positive integers, what is the value of a+ba+b?

The two roots of the quadratic equation for x x x2(ab14)x+alog2b32=0 x^2-(ab-14)x + a^{\log_2 b} - 32 = 0 are 16 16 and 22 , where 0<a<b0<a<b. What is the value of a+b a + b ?

xx and yy are real numbers that satisfy the following two equations: 3x2y=3456      and      log2(yx)=4. 3^x \cdot 2^y = 3456 \;\;\; \textrm{and} \;\;\; \log_{\sqrt{2}} (y-x) = 4. What is the value of x+y x + y ?

What is the sum of all real values of xx that satisfy 3logx2+log2x=4? 3 \log_x 2 + \log_{2} x = 4 ?


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