Algebra

# Logarithmic Functions - Solving Equations

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

Suppose that $$0 \leq x \leq 2 \pi$$ satisfies $\log_{\sin x}{\cos x} + \log_{\cos x}{\tan x} = 1.$ If $$x = \frac{a}{b}\pi$$, where $$a$$ and $$b$$ are coprime positive integers, what is the value of $$a+b$$?

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

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

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

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