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ALGEBRA III NOTES(6.7) NAME ________________ PERIOD _______________ CHANGE OF BASE FORMULA I. The change-of-base formula provides a way to evaluate logarithms of bases other than 10 or e. Change-of-base: logb x log x log b Notice that the formula changes the original logarithm into the quotient of common logs. At this point, you can use the calculator to evaluate the common logs and then simplify. Example: log4 30 log30 1.477 log 4 .602 Evaluate the following: 1) log2 14 2) log3 5 3) log4 10 4) log9 5 2.453 II. Using Logarithms to Solve Exponential Equations: One way to solve exponential equations is to use the property that if two powers with the same base are equal, then their exponents must be equal. If b x b y , then x y . Example: 43x 8x 1 Solve 2 2 3x 23 x 1 26x 23x 3 So 6x 3x 3 3x 3 x 1 However, when each side of an exponential equation cannot easily be expressed in terms of the same base, logarithms can be used to solve the exponential equation. Example: Solve 4x 7 Change to logarithmic form. log4 7 x Rewrite using change-of-base x Use the calculator to evaluate the common logs. Then simplify. Example: Solve log7 log 4 log7 .8451 1.4036 log 4 .6021 3x 1 5 Change to logarithmic form. log3 5 x 1 Rewrite using change-of-base log5 x 1 log3 Evaluate the common logs and solve for x. .6999 x 1 .4771 1.4651 x 1 .4651 x Example: Solve 32x 4 Change to logarithmic form log3 4 2x Rewrite using change-of-base log 4 2x log3 Evaluate the common logs and solve for x. .6021 2x .4771 1.2620 2x .6310 x Try this: Solve 2x 7