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## Tuesday - Algebraic and logarithmic equations

1. $(x + 3)^4 + (x+5)^4 = 16$ 
Answer: -5, -3

2. $\sqrt{x-1} + \sqrt{x+3} + 2\sqrt{(x-1)(x+3)}= 4 - 2x$ 
Answer: 1

3. $\frac{20}{\sqrt(x)} +x\sqrt(x) + x = 22$
Answer: 1, 4

4. \begin{cases}
        (\frac{x}{y})^2 + (\frac{x}{y})^3 = 12 \\
        (xy)^2 + xy = 6
   \end{cases}
Answer: (2;1), (-2;-1)

5. \begin{cases}
        (x + y)(x + 2y)(x + 3y) = 60 \\
        (y + x)(y + 2x)(y + 3x) = 105
   \end{cases}
Answer: $(\frac{19\sqrt[3](4)}{4}; \frac{-17\sqrt[3](4)}{4})$, (2; 1)

6. $x^3 + x + \sqrt[3](x^3 + x - 2) = 12$
Answer: 2

7. $6\sqrt[3](x-3) + \sqrt[3](x-2) = 5\sqrt[6]((x-2)(x-3))$ 
Answer: $\frac{190}{63}$, $\frac{2185}{728}$

8. $4^x + 10^x = 25^x$
Answwer: $log_{0.4}(\frac{-1 + \sqrt(5)}{2})$

9. Show that: $log_{3}12 = log_{3}7log_{7}5log_{5}4 + 1$

10. $\sqrt(log_{2}(2x^2)log_{4}(16x)) = log_{4}x^3$
Answer: 16

11. $4^{log_{16}x} - 3^{log_{16}x - 0.5} = 3^{log_{16}x+0.5} - 2^{2^{log_{16}x-1}}$
Answer: 64

12. $(2+\sqrt3)^{x^2-2x+1} + (2-\sqrt3)^{x^2-2x-1} = \frac{4}{2-\sqrt3}$
Answer: $1 + \sqrt2$, $1-\sqrt2$, 1

13. $(2^x - 2 * 2^{-x})^{log_{9}(2x+3)-log_{3}x} = 1$
Answer: 1, 3
