EXERCISES 4.8WRITING EXERCISES1.Explainwhyitismathematicallylegaltodefinelnxasx11tdt.For some, this type of definition is not very satisfying. Try thefollowing comparison. Clearly, it is easier to compute functionvalues forx2than for lnxand thereforex2is easier to under-stand. However, compare how you would compute (withouta calculator) function values for sinxversus function val-ues for lnx. Describe which is more “natural” and easy tounderstand.2.In this section, we discussed two different “definitions” of lnx.Explain why it is logically invalid to give different definitionsunless you can show that they define the same thing. If theydefine the same object, both definitions are equally valid andyou should use whichever definition is clearer for the task athand. Explain why, in this section, the integral definition ismore convenient than the baseelogarithm.3.Usetheintegraldefinitionoflnx(interpretedasarea)toexplainwhy it is reasonable that limx→0+lnx= −∞and limx→∞lnx= ∞.4.Use the integral definition of lnx(interpreted as area) to ex-plain why the graph of lnxis increasing and concave down forx>0.In exercises 1–4, express the number as an integral and sketchthe corresponding area.1.ln 42.ln 53.ln 8.24.ln 245.Use Simpson’s Rule withn=4 to estimate ln 4.6.Use Simpson’s Rule withn=4 to estimate ln 5.

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4-83SECTION 4.8..The Natural Logarithm as an Integral4257.Use Simpson’s Rule with (a)n=32 and (b)n=64 to estimateln 4.8.Use Simpson’s Rule with (a)n=32 and (b)n=64 to estimateln 5.In exercises 9–12, use the properties of logarithms to rewrite theexpression as a single term.9.ln√2+3 ln 2