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# Shape of a logarithmic parent graph

## Video transcript

we're asked to graph y is equal to log base 5 of X and just to remind us what this is saying this is saying that y is equal to the power that I have to raise 5 to to get to X or if I were to write this logarithmic equation as an exponential equation 5 is my base Y is the exponent that I have to raise my base 2 and then X is what I get when I raise 5 to the Y power so another way of writing this equation would be 5 to the Y power to the Y power is going to be equal to is going to be equal to X these are the same thing here we have Y as a function of X here we have X as a function of Y but they're really saying the exact same thing raised 5 to the Y power to get X this when you put as a logarithm you're saying well what power do I have to raise 5 to to get X we'll have to raise it to Y here what what do I get when I raise 5 to the Y power I get X now that out of the way let's make ourselves a little bit a little table that we can use to plot some points and then we can connect the dots to see what this curve looks like so let me pick some X's in some Y's some X's and some Y's and we in general want to pick some numbers that give us some nice round answers some nice fairly simple numbers for us to deal with so that we don't have to get a calculator and so in general you want to pick x values you want to pick X values where the power that you have to raise 5 to to get that x value is a pretty straightforward power or another way to think about it you could just think about the different Y values that you want to raise that you want to raise 5 to the power of and then you can get your X values so we could actually think about this one to come up with to come up with our actual X values but we want to be clear that when we when we express it like this the independent variable is X and the dependent variable is y we might just look at this one to pick some nice even or nice nice X's that give us nice clean answers for y so what happens so here I'm actually going to fill in the Y first just so that we get nice clean exes so let's say we're going to raise five to the let's say we're going to raise it I'm going to pick some new colors negative two negative two power and let me do some other colors negative one zero and then one I'll do one more and then two so once again this is a little bit on this little non-traditional where I'm filling in the dependent variable first but the way that we've written it over here it's actually given the dependent variable it's easy to figure out what the independent variable needs to be for this logarithmic function so what X gives me a Y of negative two what X gives me what what does X have to be for Y to be equal to negative two well 5 to the negative 2 power is going to be equal to X so 5 to the negative 2 is 1 over 25 so we get 1 over 25 so another way if we go back to this earlier one if we say log base 5 of 1 over 25 what power do I have to raise 5 to to get 1 over 25 we'll have to raise it to the negative 2 power or you could say 5 to the negative 2 is equal to 1 over 25 these are saying the exact same thing now let's do another one what happens when I raise 5 to the negative 1 power well I get one I get 1/5 so if we go to this original one over there we're just saying that log base 5 of negative sign of 1/5 want to be careful this is saying what power do we have to raise 5 to in order to get 1/5 we'll have to raise it to the negative 1 power here what power what happens when I take 5 to the 0th power I get 1 I get 1 and so this relationship this is the same thing as saying log base 5 of 1 what power to have to raise 5 to to get 1 well I just have to raise it to the 0th power let's do the next 2 what happens when I raise 5 to the first power well I get 5 so if you go look over here that's just saying log what power do I have to raise 5 to to get 5 we'll have to just raise it to the first power and then finally if I take 5 squared I get 25 five so you when you look at it from the logger and the point of view you say well what power do I have to raise 5 to to get to 25 we'll have to raise it to the second power so I kind of took the inverse of the logarithmic function I wrote it as an exponential function I switch to the dependent and independent variables so I could pick so I can derive nice clean X's that will give me nice clean y's now with that either way but I do want you to remind I could have just I I could have just picked random numbers over here but then I would have probably gotten less clean numbers over here I would have had to use a calculator the only reason why I did it this way is so I get nice clean results that I can plot by hand so let's actually graph it let's actually graph this thing over here so the Y's go between negative 2 & 2 the X's go from 125th all the way to 25 so let's let's graph it so so that is my y-axis that is my y-axis and this is my x-axis so I'll draw it like that that is my x-axis and then the Y's start at 0 then you get to positive 1 positive 2 and then you have negative 1 and you have negative 2 and then on the x axis it's all positive and I'll let you think about whether whether the domain here is well we can think about it can is this is the logarithmic function defined for an X that is that is not positive so is there any power that if that I could raise 5 to that I could get 0 no you could you could raise 5 to an infinitely negative power to get a very very very small number that approaches zero but you can never get there's no power that you can raise 5 to to get zero so X cannot be 0 and there's no power that you can raise 5 to to get a negative number so X can also not be a negative number so the domain of this function right over here and this is relevant because we want to think about what we're graphing the domain here is X has to be greater than 0 let me write that down the domain here is that X has to be greater than 0 so we're only going to be able to graph this function in the positive in the positive x-axis so with out-of-the-way X gets as large as 25 so let me graph let me put those points here so that is 5 10 15 20 and 25 and then let's plot these so the first one is in blue when X is 1 20 50 when X is 1 25th so 1 is there 1 twenty-fifths going to be really close to there then y is negative 2 so it's going to be like right over there not quite at the y-axis we're at 125th to the right of the y-axis but pretty close so that's right over there that is 1 over 25 comma negative 2 right over there then when X is 1/5 which is slightly further to the right 1/5 Y is negative 1 so right over there so this is 1/5 negative 1 and then when X is 1 Y is 0 so 1 might be right of there so this is the point 1 0 and then when X is 5 when X is 5 y is 1 when X is 5 I cover it over here but this is 5 y is 1 Y is 1 so that's the point 5 1 and then finally when X is 25 y is 2 when X is 25 Y is 2 y is 2 so this is 25 comma 2 and then I can graph the function and I'll do it let me do it in a color I haven't used this pink so as X as X gets soup or soup or soup or soup or small Y goes to negative infinity it gets really some Y to get to get X's or as X becomes you know if you say what power do you have to raise you have to raise 5 to to get you know point zero zero zero one has to be very very very negative power so Y is going to get very negative as we approach as we approach zero and then it kind of moves up like that and then starts to kind of curve to the right like that so that is and this thing right over here is going keep going down and ever in creative at a steeper and steeper rate and it's never going to [ __ ] quite touch it's never going to quite touch the y-axis it's going to get closer and closer to the y-axis but it's never going to quite touch it