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

Dec 20, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = 0.076$

#### Explanation:

Assuming the log is taken to base 10, then we can use the log rule: ${\log}_{b} \left(c\right) = {\log}_{a} \frac{c}{\log} _ b \left(b\right)$, to change our function to:
$y = \ln \frac{x}{\ln} \left(10\right) = \frac{1}{\ln} \left(10\right) \cdot \ln \left(x\right)$

$\frac{d}{\mathrm{dx}} \ln \left(x\right) = \frac{1}{x}$

$y = a \ln \left(x\right)$

$y ' = a \cdot \frac{1}{x}$

By inserting $a = \frac{1}{\ln} \left(10\right)$ we get:

$y ' = \frac{1}{x \ln \left(10\right)} = \frac{1}{\ln} \left({10}^{x}\right)$

$\frac{1}{\ln} \left({10}^{5.7}\right) = 0.076$