This defines a new functiony=logax, known as thelogarithmicfunctionwith basea, which is the exponent to whichamust beraised to get x, i.e.,x=ay⇔y=logax .Thus, the logarithmic functiony=logaxis the inverse of theexponential functiony=ax.Alogarithmofa number is anexponent:logbxis thepower to whichwe must raiseb to get x.For example,log28=3Because101

102Unit 4: Types ofFunctionsThere are two important bases:a = 10 gives rise to common logarithms, written simply as log x.a = e where e ≈ 2.71828 gives rise to natural logarithms, writtenas ln x.Common and natural logarithms may be evaluated numerically bypressing either the log orlnkeys, respectively, on a scientificcalculator.Properties of the functionf(x)=logax1.The domain of the function is the set of all positive real numbers; therange is the set of all real numbers.2.For base a > 1, f(x) is increasing. For 0 < a < 1, f(x) is decreasing.3.At x = 1, y = 0 independent of the base.The graphs of the logarithmic functionsy=log2xandy=log12xareshown in Figure on the right.These logarithmic functions may bewritten equivalently asx=2yandx=(12)y, respectively.Note that these graphs are reflections ofthe graphs ofy=2xandy=2−x,respectively, in the liney=x.Examples1.Evaluate the following:a)log864,b)log3181,c)log162.Solution.

a)Lety=log864, then8y=64=82and soy=2.b)Let y =log3181,then3y=181=3−4and soy=−4.c)Let y =log162, then16y=2or(24)y=24y=2and so 4y = 1 and thereforey=14.2.Solve the following for x:a)log4x=3,b)log81x=34Solution.a)x=43=64.b)x=8134=(8114)3=33=27.NotationThe notationlog(x)is understood to belog10(x); for example,log(7) meanslog10(7). The notationln(x)is understood to beloge(x); for example,ln(7)meansloge(7). Check on yourcalculator thatlog 12=1.0791812and¿12=2.4849066.Examples1.Solve the equationln(x+4)2=3for x.Solution.2ln(x+4)=3103

104Unit 4: Types ofFunctionsln(x+4)=32x+4=e1.5x+4=4.48169to 5 decimal placesx=0.48169to 5 decimal places2.Expressloga3+loga4−loga6as a single logarithm.Solution.6=loga(3×4)−¿loga6loga3+loga4−loga¿¿loga(3×46)¿loga23.Find the value ofxsatisfyinglogax=3loga2+loga20−loga1.6Solution.logax=3loga2+loga20−loga1.6¿loga23+loga20−loga1.6¿lo ga(8×201.6)¿loga100Thereforex=100.Unit 4: Types of Functions4.4 The Exponential andLogarithmic Functions

Examples4.Solve the equationln(x+4)2=3for x.Solution.2ln(x+4)=3ln(x+4)=32x+4=e1.5x+4=4.48169to 5 decimal placesx=0.48169to 5 decimal places5.Expressloga3+loga4−loga6as a single logarithm.Solution.6=loga(3×4)−¿loga6loga3+loga4−loga¿¿loga(3×46)¿loga26.Find the value ofxsatisfyinglogax=3loga2+loga20−loga1.6Solution.logax=3loga2+loga20−loga1.6¿loga23+loga20−loga1.6¿lo ga(8×201.6)¿loga100105

106Unit 4: Types ofFunctionsThereforex=100.Unit 5: Mathematics of Finance5.0 Simple and CompoundInterest CalculationsToday, businesses and individuals are faced with a bewildering array ofloan facilities and investment opportunities. In this section we explainhow these financial calculations are carried out to enable an informedchoice to be made between the various possibilities available.

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Rational number, Augustus De Morgan, Set Operations