Question

What is the slope of y = log(x) when x=20 ?


The formula for the slope is _________ for h close to 0 (but not equal )

The best estimate for the slope is____

Answer 1

${ \qquad\qquad\huge\underline{{\sf Answer}}}$

The formula for slope according to first principle is :

$\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_{x\to0} \: \: \sf \frac{f(x + h) - f(x)}{h} \]$

$\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_{x\to0} \: \: \sf \frac{ log(x + h) - log(x)}{h} \]$

$\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_{x\to0} \: \: \sf \frac{ log( \frac{x + h}{x} ) }{h} \]$

$\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_{x\to0} \: \: \sf \frac{ log( \frac{ h}{x} + 1 ) }{h} \]$

$\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_{x\to0} \: \: \sf \frac{ log( \frac{ h}{x} + 1 ) }{ \frac{h}{x} \times x} \]$

$\qquad \sf  \dashrightarrow \: \[ \displaystyle\lim_{x\to0} \: \: \sf \frac{1}{x} \bigg(\frac{ log_{e} ( \frac{ h}{x} + 1 ) }{ \frac{h}{x} \times log_{e}(10) } \] \bigg)$

$\qquad \sf  \dashrightarrow \: \[ \dfrac{1}{x \: log_{e}(10) } \displaystyle\lim_{x\to0} \: \: \sf \bigg(\frac{ log_{e} ( \frac{ h}{x} + 1 ) }{ \frac{h}{x} } \] \bigg)$

$\qquad \sf  \dashrightarrow \: \[ \dfrac{1}{x \: log_{e}(10) }$

Now, the best estimate of slope is at x = 20 :

$\qquad \sf  \dashrightarrow \: \[ \dfrac{1}{20\times 2.303}$

$\qquad \sf  \dashrightarrow \: \[ \dfrac{1}{46.06}$

$\qquad \sf  \dashrightarrow \: \approx0.0217$