$($a$)$ Write a Python function which has the single argument $n$ and returns

the sum of the natural logarithms of the first $n$ natural numbers.

$($b$)$ Use the function you wrote in part $($a$)$ to calculate the sum of the

natural logarithms of the first $100$ natural numbers.

$($c$)$ Write a Python function which returns the minimum value of $n$ such

that

${\displaystyle \underset{i=1}{\overset{n}{}}}ln(i)T$

for some given number $T.$

$($d$)$ Use the function you wrote in part $($c$)$ to find the minimum number of

terms needed such that the sum of the logarithms is greater than $100.$

$($e$)$ A student has written the following code to calculate the real roots of

the quadratic equation $a{x}^2+bx+c=0,$ where $a,b$ and $c$ are real, or

return a message if there are none.

def solve$\_$quadratic $($a$,$b$,$c$)$:

disc $=$ b$^2-4$c

if disc $<mtext></mtext><mn>0</mn>$ then:

return There are no real solutions to your equation

else

solsmall $=(-$b$-$sqrt$($disc$))/(2/$a$)$

sollarge $=(-$b$+$sqrt$($disc$))/(2/$a$)$

return solsmall, solarge

Explain the errors in this code, and correct them to produce a working

function. Use your corrected code to find the real roots, if any, of

$4x^2+19x-5=0$ and $4x^2+19x+100=0$