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Hello again. Were still looking at logarhythms, and this time were looking at a very important rule, an idea of kind of like cancelling it out. Weve looked at that in previous modules. Cancelling to like, get a variable say, on its own. In this case, what we have is log to the base 4 of 4 to the 8. Well, the nice thing here is, what is a logarhythms? Well, its asking us to figure out what does the base have to be put to in order to get the term in say the brackets I just put in. so what does 4 have to be raised to, or put -- what power does it have to be put to become 4 to the 8? Well, if we think about it, thats kind of there in the answer. What does 4 have to be put through to get to be 4 to the power of 8? Well, 8. So this can merely be thought of as 8. And you think, if you have the base and the number here are the same, all thats left is the exponent. In fact, we can apply that power rule and rewrite this as 8 log 4 of 4. Well what does 4 need to be raised to, to get 1? O

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0:20 1:36 How To Evaluate when e is Raised to the Power of a Natural Logarithm YouTube Start of suggested clip End of suggested clip But then if i have ln of e raised to an x. That will then therefore its just going to equalMoreBut then if i have ln of e raised to an x. That will then therefore its just going to equal whatever my e what power its raised to but that is also true if i have e raised to the ln.

The logarithms and exponentials cancel each other out (equation (4)), giving our product rule for logarithms, ln(xy)=ln(x)+ln(y).

These equations simply state that ex and lnx are inverse functions. Well use equations (3) and (4) to derive the following rules for the logarithm.Basic rules for logarithms. Rule or special caseFormulaQuotientln(x/y)=ln(x)ln(y)Log of powerln(xy)=yln(x)Log of eln(e)=1Log of oneln(1)=02 more rows

Although the base of a logarithm can be any number, the most common bases used in science are 10 and e, which is an irrational number known as Eulers number. To distinguish them, mathematicians use log when the base is 10 and ln when the base is e.

0:08 2:46 How do you solve an exponential equation with e as the base YouTube Start of suggested clip End of suggested clip Form this is now going to be log base e of 12 equals 3x. Now remember we write log base e of 12 asMoreForm this is now going to be log base e of 12 equals 3x. Now remember we write log base e of 12 as ln of 12 equals 3x. So now to solve for x you divide by 3.. So x equals ln base 12 of 3.

2:17 10:15 The Exponential Function e and The Natural Log ln - YouTube YouTube Start of suggested clip End of suggested clip So we get rid of e by applying ln to both sides. So ln e if you have x equals ln five.MoreSo we get rid of e by applying ln to both sides. So ln e if you have x equals ln five.

Put in the base number e on both sides of the equation. e and ln cancel each other out leaving us with a quadratic equation.

The cancellation formulas for logs are: loga(ax)=x, for every xR, aloga(x)=x, for every x0.

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