Irrational and transcendental equations

Radical equation

$$\sqrt[n]{x}=a$$

Solution:

$$x=a^n$$

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Exponential equation

$$a^x=b$$

Solution:

$$x={\log }_{a}b=\frac{lg b}{lg a}$$

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Logarithmic equation

$${\log }_{a}x=b$$

Solution:

$$x=a^b=e^{b·\ln a}={10}^{b·lg a}$$

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Trigonometric equations

$$\sin x=a$$

Solution:

$$\begin{gathered} x_1=arc \sin a+2k\mathrm{\pi } \\ {\mathrm{x}}_2=\mathrm{\pi }-{\mathrm{x}}_1 \end{gathered}$$

$$\cos x=a$$

Solution:

$$\begin{gathered} x_1=arc \cos a+2k\mathrm{\pi } \\ {\mathrm{x}}_2=\mathrm{\pi }-{\mathrm{x}}_1 \end{gathered}$$

$$tg x=a$$

Solution:

$$x_1=arc tg a+k\mathrm{\pi }$$

$$ctg x=a$$

Solution:

$$x_1=arc ctg a+k\mathrm{\pi }$$

k is an arbitrary integer