Solving Certain Logarithmic Equations - M. Olsen's Website
In this section we will discuss Graphing logarithmic functions. Equivalent EquationsSolving some Logarithmic EquationsWhat is a logarithmic function?It is the inverse of an exponential function!!Consider y=2x Is this a function? ______ Is it a 1-1 function? _______ So will the inverse be a function? __________382968595250to find the inverse we __________________ the ____ & ____ variables.Switching the x & y variables we getx=2y Label each equation which is graphed on the right:fx=2xf-1(x)y=xThis is really what we are calling the log base 2. This says the base number (2) raised to the y value gives you the number x.Note: log2x is read “the log base 2 of x” and means “the exponent to which we raise 2 to get x.”The Meaning of logaxFor x > 0 and a a positive constant other than 1, logax is the exponent to which a must be raised in order to get x. Thus, logax = m means am = xor equivalently, logax is that unique exponent for whichSo if 23=8Then log28=3log2x=y is just a notation we use to say 2y=x. And it is the inverse of fx=2xMeaning:2701977162748a) 49 = 7x is equivalent to x = log749 b) x-2 = 9 is equivalent to –2 = logx9. The base remains the base.The exponent is the logarithm.Ex: Rewrite each as an equivalent logarithmic equation:a) 49=7x b) x-2=9Quickly, on the same graph sketch the graphs of fx=2x;gx=3x;h(x)=5x.32157021634Now sketch the graphs of f-1x=log2x;g-1x=log3x;h-1x=log5x on the same graph above.Ex: Label some possible functions that these graphs could represent if they were sketched.Solving Certain Logarithmic EquationsEx: Solve the following: a) log5125=y b) log3x=4 c) Log2x=-3logaaThe logarithm, base a, of a is always 1: logaa = 1. loga1The logarithm, base a, of 1 is always 0: loga1 = 0. Ex:log22=____=log33 & log21=____=log31=log51 The Principle of Exponential EqualityFor any real number b, where(Powers of the same base are equal if and only if the exponents are equal.)Ex: Solve:a) log636=x b) log55=xc) log8x=23x=82/3x=382x=22=45=5x51=5x1=x36=6x62=6x2=x ................
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