How to Graph x/ln(x)

SOLVED:a. Graph the five functions lnx-2, lnx-1, lnx, lnx+1, and lnx+2 on the window [0,4] by [-3,3]. b. Find the slope (using NDERIV or d y / d x ) of several of the curves at a particular x -value and check that in each case the slope is the reciprocal of the x -value. This suggests that the derivative of each function is 1 / x. c. Based on part (b), conjecture what is the indefinite integral of the function 1 / x (for x>0 ).

SOLVED:The natural logarithm function, ln, is defined so that e^lnx=x for any positive number x (a) Sketch a graph of the natural logarithm function. (b) Prove the identities lna b=lna lnb and lna^b=b lna (c) Prove that (d)/(d x) lnx=(1)/(x) (d) Derive the useful approximation ln(1 x) ≈x, which is valid when |x| ≪1 . Use a calculator to check the accuracy of this approximation for x=0.1 and x=0.01.

Introduction to Logarithms – Explanation & Examples

Natural Logarithm | Definition, Addition & Antiderivative