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In this lesson, we explore how to graph the function x/ln(x). We carefully explain how this function has a restricted domain and some fascinating features whose complete description involves limits and derivatives.
1.2K viewsOct 17, 2023
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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.
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.
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