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$$2^x = 2^4 \implies x = 4$$

The value of $x$ is $\boxed{\mathbf{4}}$.

42. Calculate the value of $y$ if $3^{y} = \frac{1}{27}$.

$$3^y = 3^{-3} \implies y = -3$$

The value of $y$ is $\boxed{\mathbf{-3}}$.

43. Calculate the value of $z$ if $5^{z} = \sqrt{125}$.

$$5^z = 5^{\frac{3}{2}} \implies z = \frac{3}{2}$$

The value of $z$ is $\boxed{\mathbf{\frac{3}{2}}}$.

44. Calculate the value of $p$ if $4^{p} = 8$.

$$4 = 2^2$$ $$8 = 2^3$$

$$2p = 3 \implies p = \frac{3}{2}$$

The value of $p$ is $\boxed{\mathbf{\frac{3}{2}}}$.

45. Calculate the value of $q$ if $27^{q} = 9$.

$$27 = 3^3$$ $$9 = 3^2$$

$$3q = 2 \implies q = \frac{2}{3}$$

The value of $q$ is $\boxed{\mathbf{\frac{2}{3}}}$.

46. Calculate the value of $r$ if $1000^{r} = 100$.

$$1000 = 10^3$$ $$100 = 10^2$$

$$3r = 2 \implies r = \frac{2}{3}$$

The value of $r$ is $\boxed{\mathbf{\frac{2}{3}}}$.

47. Calculate the value of $s$ if $64^{s} = 4$.

$$64 = 4^3$$ $$4 = 4^1$$

$$3s = 1 \implies s = \frac{1}{3}$$

The value of $s$ is $\boxed{\mathbf{\frac{1}{3}}}$.

48. Calculate the value of $t$ if $81^{t} = 3$.

$$81 = 3^4$$ $$3 = 3^1$$

$$4t = 1 \implies t = \frac{1}{4}$$

The value of $t$ is $\boxed{\mathbf{\frac{1}{4}}}$.

49. Calculate the value of $u$ if $125^{u} = 5$.

$$125 = 5^3$$ $$5 = 5^1$$

$$3u = 1 \implies u = \frac{1}{3}$$

The value of $u$ is $\boxed{\mathbf{\frac{1}{3}}}$.

50. Calculate the value of $v$ if $16^{v} = 2$.

$$16 = 2^4$$ $$2 = 2^1$$

$$4v = 1 \implies v = \frac{1}{4}$$

The value of $v$ is $\boxed{\mathbf{\frac{1}{4}}}$.