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# Solving exponential equations using logarithms: base-2

## Video transcript

let's say that we've got the function y is equal to five times two to the T power and someone were to come up to you and say hey look you know this is an interesting function but I I'm curious I like the number 1111 and I'm curious at what point for what T value will my Y be equal to 1111 and so I encourage you to pause this video and think about it on your own at what T value will this will Y be equal to a roughly equal to 1111 and if you see the need you might want to use a calculator so I'm assuming you've given a go at it let's work through this together so we want to say when does 5 times 2 to the T power equal 1111 so let's write that down so when does 5 times 2 to the T power equal 1111 so whenever we're doing anything algebraically it's always a little bit useful to see if we can isolate the variable that we're trying to solve for we're trying to find what T value will make this equal that right over there so a good first step would maybe try to get this 5 out of the left hand side so let's divide the left by 5 but if we want to keep this to being in equality we have to do the same thing to both sides so we get 2 to the T power 2 to the T power is equal to 1111 over 5 so how do we solve for T here well what function is essentially the inverse of the exponential function what would be the logarithm if we say if we say that a to the B power is equal to C then that means that log base C I'm sorry log base a of C is equal to V a to the B power is equal to C log base a of C says what power do I need to raise a to to get to see while you to raise a to the B power to get to see a to the B power is equal to C so these two are actually equivalent statements so let's take log base two of both sides of this equation so on the left-hand side on the left-hand side you have log base 2 of 2 to the T power 2 to the T power and on the right hand side you have log base 2 of 1111 over 5 now why is this useful right over here so this is what power do we have to raise 2 to to get to 2 to the T power well to get to 2 to the T power we have to raise 2 to the to the T power so this thing right over here this thing right over here just simplifies to this just simplifies to T that just simplifies to T and on the right hand side we have log base 2 we have all of this business right over here so I'll just write it over T is equal to log base 2 of 1111 over 5 so this is an expression that gives us our T value but then the next question is well how do we actually figure out what this is and if you take out your calculator you will quickly notice that there is no there is no log base 2 button so how do we actually compute it and here we just have to apply a very useful property of exponents if we have if we have log if we have log base 2 of well really anything well let me write it this way if we have log base a of C we can compute this as log base anything of C over log base that same anything of a this anything has to be the same thing and our calculator is useful because it has a log the log when you just press log it's log base 10 if you press Ln its natural log or log base e I like to just use the log base 10 so this is going to be the same thing as log base 10 of 1111 over 5 over log base 10 of 2 so we can get our calculator out and we could have done log base E if we wanted that would've been natural log but I'll just use a log button so this is logarithm of 1111 over 5 so that's this part right over here this is log base 10 implicitly that's what the log button is divided by divided by log base 10 of 2 and then that gives us 7 well it just keeps on going this is approximately equal to 7 seven point seven nine six well just seven point seven nine six so this is approximately equal to seven point seven nine six so when T is roughly equal to that you're going to have Y equaling 1111