Question

Given the function `y=log(x), 0<x<10`, what is the slope of the graph where `x=5.7`?

Answer 1

`(dy)/(dx)=0.076`

Explanation:

Assuming the log is taken to base 10, then we can use the log rule: `log_b(c)=log_a(c)/log_b(b)`, to change our function to:
`y=ln(x)/ln(10)=1/ln(10)*ln(x)`

`d/(dx)ln(x)=1/x`

`y=aln(x)`

`y'=a*1/x`

By inserting `a=1/ln(10)` we get:

`y'=1/(xln(10))=1/ln(10^x)`

`1/ln(10^5.7)=0.076`