What is ((ln(1/(x+h)+1)-ln(1/x+1)))/h ?

The solution to ((ln(1/(x+h)+1)-ln(1/x+1)))/h is (ln((x(1+x+h))/((x+h)(1+x))))/h

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simplify ((ln(1/(x+h)+1)-ln(1/x+1)))/h

simplify

\frac{( ln ( \frac{1}{x + h} + 1 ) - ln ( \frac{1}{x} + 1 ) )}{h}

\frac{ln ( \frac{x ( 1 + x + h )}{( x + h ) ( 1 + x )} )}{h}

Solution steps

\frac{( ln ( \frac{1}{x + h} + 1 ) - ln ( \frac{1}{x} + 1 ) )}{h}

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (\frac{1}{x + h} + 1) - ln (\frac{1}{x} + 1) = ln (\frac{\frac{1}{x + h} + 1}{\frac{1}{x} + 1})

= \frac{ln ( \frac{\frac{1}{x + h} + 1}{\frac{1}{x} + 1} )}{h}

Simplify

\frac{\frac{1}{x + h} + 1}{\frac{1}{x} + 1} : \frac{x ( 1 + x + h )}{( x + h ) ( 1 + x )}

= \frac{ln ( \frac{x ( 1 + x + h )}{( x + h ) ( 1 + x )} )}{h}

ln (\frac{413.15}{293.15})

- \frac{ln ( 18 )}{3 ln ( 2 ) - ln ( 3 )}

\frac{( ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ )}{2}

lo g_{e} (\frac{8}{3})

7.52 + lo g_{10} (\frac{0.02}{0.48})

What is ((ln(1/(x+h)+1)-ln(1/x+1)))/h ?
The solution to ((ln(1/(x+h)+1)-ln(1/x+1)))/h is (ln((x(1+x+h))/((x+h)(1+x))))/h