Use the change of base formula for logarithmic functions and incorporate the fact that $\displaystyle \ln 1=0$ and $\displaystyle e^{\ln }=1$

$\displaystyle \log_ax=\frac{\log_bx}{\log_ba}$

Or

$\displaystyle \log_ax=y$ can be solved using $\displaystyle a^y=x$

$\displaystyle \ln 1=x$

$\displaystyle e^{x}=1$

$\displaystyle x=0$

Imaginary numbers do not fall on the number line-- they are by definition not real numbers. 

** If the question asked where $\displaystyle 4i^{2}$ falls on the number line, the answer would be to the left of 0, because $\displaystyle 4i^{2}=-4$.

To see what the polar form of the number is, it helps to draw it on a graph, where the horizontal axis is the imaginary part and the vertical axis the real part. This is called the complex plane.

To find the angle $\displaystyle \theta$, we can find its supplementary angle $\displaystyle \Psi$ and subtract it from $\displaystyle \pi$ radians, so $\displaystyle \theta = \pi - \Psi$.

Using trigonometric ratios, $\displaystyle \tan \Psi = \frac{1}{\sqrt{3}}$   and  $\displaystyle \arctan({\frac{1}{\sqrt{3}}})= \Psi = \frac{\pi}{6}$.

Then $\displaystyle \theta = \frac{5\pi}{6}$.

To find the distance $\displaystyle r$, we need to find the distance from the origin to the point $\displaystyle (-\sqrt{3}, 1)$. Using the Pythagorean Theorem to find the hypotenuse $\displaystyle r$, $\displaystyle r = \sqrt{ \sqrt{3}^2 + 1^2 }$ or $\displaystyle r=2$.

 Imaginary numbers do not fall on the number line by definition, since they are not real numbers. However, although i is an imaginary number equal to the square root of -1, $\displaystyle i^2$ is a real number since $\displaystyle i^2 = -1$. Therefore, $\displaystyle 5i^2 = -5$. Negative numbers fall to the left of 0 on a number line.

In complex numbers of the form $\displaystyle a + bi$, a represents the real portion of the number and b represents the imaginary portion of the number. To graph $\displaystyle a + bi$ on a plane in which real numbers are represented by the x-axis and imaginary numbers are represented by the y-axis, place a point a units right of the origin and b units above the origin. The graph shows a point 2 units right and 3 units above the origin, so the complex number represented is $\displaystyle 2 + 3i$.

In complex numbers of the form $\displaystyle a+bi$, a represents the real portion of the number and b represents the imaginary portion of the number. To graph $\displaystyle a + bi$ on a plane in which real numbers are represented by the x-axis and imaginary numbers are represented by the y-axis, place a point a units right of the origin and b units above the origin. The graph shows a point 5 units left and 2 units below the origin, so the complex number represented is $\displaystyle -5 - 2i$.

 In complex numbers of the form $\displaystyle a + bi$, a represents the real portion of the number and b represents the imaginary portion of the number. In the complex number $\displaystyle 4 - 7i$, $\displaystyle a = 4$ and $\displaystyle b = -7$.

 In complex numbers of the form $\displaystyle a + bi$, $\displaystyle a$ represents the real portion of the number and $\displaystyle b$ represents the imaginary portion of the number. In the complex number $\displaystyle -3 + ki$, $\displaystyle a = -3$ and $\displaystyle b = k.$

 In complex numbers of the form $\displaystyle a + bi$, a represents the real portion of the number and b represents the imaginary portion of the number. To graph $\displaystyle a + bi$ on a plane in which real numbers are represented by the x-axis and imaginary numbers are represented by the y-axis, place a point a units right of the origin and b units above the origin. The graph shows a point 4 units right and 7 units below the origin, so the complex number represented is $\displaystyle 4 - 7i$.

In complex numbers of the form $\displaystyle a + bi$, a represents the real portion of the number and b represents the imaginary portion of the number. To graph $\displaystyle a + bi$ on a plane in which real numbers are represented by the x-axis and imaginary numbers are represented by the y-axis, place a point a units right of the origin and b units above the origin. The graph shows a point 8 units left and 1 unit above the origin, so the complex number represented is $\displaystyle -8 + 1i = -8 + i$.