448 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 6.3 EXPONENTIAL EQUATIONS AND INEQUALITIES In this section we will develop techniques for solving equations involving exponential functions. Suppose, for instance, we wanted to solve the equation 2\"" = 128. After a moment's calculation, we nd 128 = 27, so we have 21 = 2?. The onetoone property of exponential functions, detailed in Theorem 6.4, tells us that. 23 2 2T if and only ifcc = 7. This means that. not only is at = 7 a solution to 2I = 27, it is the only solution. Now suppose we change the problem ever so slightly to 2I = 129. We could use one of the inverse properties of exponentials and logarithms listed in Theorem 6.3 to write 129 = 2103;2(129). W'e'd then have 2I = 2103;2(129), which means our solution is x = log2(129). This makes sense because, after all, the denition of log2(129) is 'the exponent we put on 2 to get 129.1 Indeed we could have obtained this solution directly by rewriting the equation 21 : 129 in its logarithmic form log2(129) = 3:. Either way, in order to get a reasonable decimal approximation to this number, we'd use the change of base formula, Theorem 6.7, to give us something more calculator friendly,l say log2(129) = %. Another way to arrive at this answer is as follows 2x = 129 ln (23) = ln(129) Take the natural log of both sides. :cln(2) = ln(129) Power Rule l a: : n(129) ln(2) "Taking the natural log\" of both sides is akin to squaring both sides: since f('.) = ln(:'c) is a function, as long as two quantities are equal, their natural logs are equal.2 Also note that. we treat ln(2) as any other nonzero real number and divide it through3 to isolate the variable .13. we summarize below the two common ways to solve exponential equations, motivated by our examples. Steps for Solving an Equation involving Exponential Functions 1. Isolate the exponential function. 2. (a) If convenient, express both sides with a common base and equate the exponents. (1)) Otherwise, take the natural log of both sides of the equation and use the Power Rule. Example 6.3.1. Solve the following equations. Check your answer graphically using a calculator. 1- 233 = 16\" 2. 2000 = 1000 - 3-0-115 3_ 9 . 3x 2 72m 4- 75 = iii 5. 25$ = 53 + 6 6. EL?" 2 5 Solution. LYou can use natural logs or common logs. We choose natural logs. (In Calculus, you'll learn these are the most. 'mathy' of the logarithms.) 2This is also the 'if' part. of the statement. logbr.) = logbfw) if and only if u. = w in Theorem 6.4. 3Please resist the temptation to divide both sides by 'ln' instead of ln(2]. Just like it wouldn't make sense to divide both sides by the square root symbol Rf: when solving cox/2 = 5, it. makes no sense to divide by 'ln'. h (a) Find the inuersere of the function f(:[:) : 3 I: . (b) Solue the equation 3:3 : 100, showing each step. Explain how the inverse of the function f comes into play in solving the equation. (0) How does our text solve the equation 23" = 129 on page 448? In view of part (b), explain why it does not make sense to solve the equation this way. (a) Solve the equation 233 = 129 using the idea of an inverse function. Eaplain and be sure to show each step