Good day students welcome to mathgotserved.com in this clip were going to be going over how to solve exponential equations rights the instructions for problems 123 are for us to solve the exponential equations for the exact solutions alright so let's go see there problem one let's say we have five race to the X power equals X hundred and 25 first question will going to ask yourself since we are looking for the exact solution is that can I express the number on the right as a power rising exponents with the same base asked the exponential term to the left so that's the question can I write 625 is five to certain power so let's go ahead then not go over the multiply the powers of five and see if we run across 625 so the first one five to the first power is 55 to the second power 5×5 is 25 five to the third power is 125×500+25 which is 125 and then we have five to the fourth power 125×5 you because of multiplication you end up with 625 and that's exactly what's we are looking for now will proceed to express both sides of X of exponents with the base of five so have five to the X power on the left side equals five to the fourth power on the right side now what you can do is simply Council of the two bases and equation the exponents but let me show you what justifies that will what you going to do is basically take log base five of both sides of the equation so this format of expressing the next step shows you the inverse relationship between exponents the logs remember that exponents are lot and logs are inverses so these ones counsel out so we have X equals on the right side log base five and and exponents of the base of five the log pics obviate the exponents base and you left with the power so our final answer is X equals four now let's take a look at question number two what if we have one third 3X power equals 80 one okay now the goal is Om to try to express both sides of exponents with the same base if I can do that the Nike is log with him base that base two counsel out the base is an equation the exponents to gets my final result now let's take a look at the left side we have one third I can express one third using the reciprocal property of exponents as three to the negative one okay and raise after the X power and then we have 80 one on the right side now using the power power property if you have a power race to the power all you simply do is multiply the powers so we have negative one times X of the simplification of the left side of this equation to have three to the X equals 80 one now can I express 80 one of the power all 3 | powers of three okay so three to the first power is three that's not forget three to the zero power is one that's important in some problems three to the second power is nine we to the third power is 27 and three to the fourth power 27×3 is 80 one this is the one we're looking for some of them I rewrite this equation as powers of three on both sides so three to the X is equal to three to the fourth power and then if we take log base three just to show you to steps you have to do this out take log base three of both sides of the equation the shows you what's can happen next figure log base three of both sides of the equation eliminates the bases of the exponential terms that we have on both sides like that it will left Om with OMF let's something else we have Om if you multiply BC we have a negative here so let's the forget negative but a backend okay so we can a have negative X is equal to four right now Om let's finish this off to get X isolated we divide both sides you negative one and our final answer is going to be X equals negative for RIT so that's the Om solution to question number two now let's take a look at question three what if we have equation 5 to the 2X plus one equals 125 now we are ready have five of the power here five is prime so we can do anything more with this but can we express 125 as a power apply and the answer is yes if you go back to the other list of powers of five that we have you find out that five race to the third power is 125 now we have exponents with the same base on both sides which is excellent so what it take log base five of the left side of the equation and log base five of the right side of the equation now shows the reason why we can counsel out the base is Om in an exponential equation where the bases of terms of both sides are identical right so we have they a consolation happening here cancel cancel because your inverses and we left with 2X plus one equals three now to finish this off would you subtract one and divide by two let's subtract one from both sides that yields 2X equals two and then will divide both sides by two we get our final results X equals one of the solution to our exponential equation okay now for problems 45 and six we going to be sold in exponential equations with the aid of our calculators okay so the instructions are for us to solve and round your
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