Evaluating natural logarithm with calculator

use a calculator to find log base e of 67 to the nearest thousand so just as a reminder E is one of these crazy numbers that shows up in nature and Finance and all these things and it's approximately equal to two point seven one and it just keeps going on and on and on so you could view log base E as 67 you might say what does e mean e is just a number just like pi is just a number so this is really the same thing as saying log base two point seven one and the actual number so you'd have to write all the digits that keep on going forever and and never repeat six of 67 so what power do I have to raise e to to get to 67 so another way of saying that is if this is equal to X you're saying e to the X is equal to 67 we need to figure out what X is now traditionally you will never see someone write log base E even though E is one of the most common basis to take a logarithm of and so the reason why you wouldn't see log base E written this way is log base E is referred to as the natural logarithm and I think that's used because I shows up so many times in nature so log base E of 67 another way of saying that or seeing that and the more typical way of seeing that is the natural log the lot and I think that this is Ln so I think it's maybe from French or something log natural of 67 so this is the same thing as log base e of 67 this is saying the exact same thing to what power do I have to raise e to to get 67 when you see this Ln it literally means log base E now they let us use a calculator and that's good because I don't know off the top of my head what power up to raise to point seven one so on and so forth to what power have to raise that to to get to 67 so we'll get our calculator out so we get the ti-85 out and different calculators will have different ways of doing it if you have a graphing calculator like this you literally can literally type in the statement natural log of 67 and then evaluate it so here this is the button for Ln means natural log log natural maybe Ln of 67 and then press ENTER give you the answer if you don't have a graphing calculator you might have to press 67 or and then a value and then press natural log to give you the answer but a graphing calculator can literally type it in the way that you would write it out and then you press Enter so for point two zero four six nine and we want to round to the nearest thousandth so this is a thousands place right here this for the digit after that is five or larger to six so we're going to round up so this is four point two zero five so this is approximately equal to four point two four point two zero five and it actually makes a lot of sense because we know that e is greater than 2 and it is less than and it is less than three and if you think about what 2 to the fourth power two to the fourth power gets you to 16 and 3 to the fourth power gets you to get you to 81 67 is between 16 and 81 and E is between 2 and 3 so it at least it feels right that's something that's like two point seven one two the little over the fourth power should get you to a number that's pretty close to three to the fourth power and actually that makes sense because it's actually closer to three two point seven one is closer to three than it is to two so this feels right that you take this to the fourth little over the fourth power you get to 67