Chapter solving exponential and logarithmic functions

SAT Problem of the day

 Solve simple and more complex exponential and logarithmic functions objectives

 Remember that exponential and logarithmic functions are one-to- one functions. That means that they have inverses. Also recall that when inverses are composed with each other, they inverse out and only the argument is returned. We're going to use that to our benefit to help solve logarithmic and exponential equations.  Please recall the following facts:  log a a x = x  log 10 x = x  ln e x = x  a log a x = x  10 log x = x  e ln x = x Exponential and logarithmic functions

 Isolate the exponential expression on one side.  Take the logarithm of both sides. The base for the logarithm should be the same as the base in the exponential expression. Alternatively, if you are only interested in a decimal approximation, you may take the natural log or common log of both sides (in effect using the change of base formula)  Solve for the variable.  Check your answer. It may be possible to get answers which don't check. Usually, the answer will involve complex numbers when this happens, because the domain of an exponential function is all reals. Solving exponential equations

If x = 1, then 3 1 = 3. If x = 2, then 3 2 = 9. If x = 3, then 3 3 = 27. Solve this by trying some small values of x (i.e. try x = 1, 2, etc. ) Since 3 3 = 27, x = 3 is the solution to the equation 3 x = 27. Summarize the solution. Example#1 Solve the equation 3 x =27.

e 2x = 35 / 3. Divide both sides by 3 in order to prepare this equation for the application of the natural log function. ln e 2x = ln (35 / 3 ). Apply the natural log function to both sides. 2x = ln ( 35 / 3 ) Use the inverse property of logs and exponents. x =(1/2) ln (35 / 3 )Solve for x Use a calculator to find a 4 decimal approximation to the answer. Example#2 Solve the equation 3 e 2x = 35 for x and use a calculator to give a 4 decimal place approximation answer..

e 3x + 2 = 20 Subtract 3 from both sides of the equation in order to prepare it for the application of the natural log function. ln e 3x + 2 = ln (20). Apply the natural log function to both sides. 3x + 2 = ln ( 20 ) Use the inverse property of logs and exponents. x =(1/3) ( -2 + ln 20 )Solve for x Use a calculator to find a 4 decimal approximation to the answer. Example#3 Solve the equation 3 + e 3x + 2 = 23 for x and use a calculator to give a 4 decimal place approximation answer.

 Solve for x : 4·5 2x = 64. Solution: Example#4 4·5 2x = x = 16 log 5 5 2x = log x = log x = 2x x 0.861

 Do problems 1-6 in your worksheet Student guided practice

 Use properties of logarithms to combine the sum, difference, and/or constant multiples of logarithms into a single logarithm.  Apply an exponential function to both sides. The base used in the exponential function should be the same as the base in the logarithmic function. Another way of performing this task is to write the logarithmic equation in exponential form.  Solve for the variable.  Check your answer. It may be possible to introduce extraneous solutions. Make sure that when you plug your answer back into the arguments of the logarithms in the original equation, that the arguments are all positive. Remember, you can only take the log of a positive number. Logarithmic Equations

 Solve for x : log 3 (3x) + log 3 (x - 2) = 2. log 3 (3x) + log 3 (x - 2) = 2 log 3 (3x(x - 2)) = = 3x(x - 2) 9 = 3x 2 - 6x 3x 2 - 6x - 9 = 0 3(x 2 - 2x - 3) = 0 3(x - 3)(x + 1) = 0 x = 3, - 1 Example#5

 ln (x + 4) + ln (x - 2) = ln 7  First we use property 1 of logarithms to combine the terms on the left.  ln (x + 4)(x - 2) = ln 7  Now apply the exponential function to both sides.  e ln (x + 4)(x - 2) = e ln 7  The logarithmic identity 2 allows us to simplify both sides.  (x + 4)(x - 2) = 7  x 2 + 2x - 8 = 7  x 2 + 2x - 15 = 0  (x - 3)(x + 5) = 0  x = 3 or x = -5  x = 3 checks, for ln 7 + ln 1 = ln 7. Example#5

 solve −6log 3 (x − 3) = −24 for x Example#6

 log 9 (x + 6) − log 9x = log 92 Example#7

 Do problems 1-8 in your worksheet Do problems

 Let’s watch solving exponential and logarithmic function Video

 Do problems in your book page 217 Homework

 Today we learned to solve exponential and logarithmic functions  Next class we are going to learn about exponential and logarithmic models closure

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