Solving Exponential and Logarithmic Equations 

Professor Dave Explains there is also one mistake at the very end of the video at 6:23 for ln(x+1)^2=2. There should be two answers, which is -1 plus or minus e. The negative sign comes from the one when you take the square root of both sides. When you plug in -1-e in the original equation, squaring the negative value will still yield positive answer. For example, (-1)^2=1 and 1^2=1. In case you didn’t see the properties of logarithms of the even powers, it would be ln(x+1)^2=2ln(abs(x+1)). abs stands for absolute value. The absolute value is there because the value can be negative at that value. If there’s no absolute value there, it would make the answer undefined.

log₂(-1) = log₂(-1×1)=log₂(-1)+log₂(1) log₂(-1)+log₂(-4)=log₂(-1×-4)=log₂(4)=2 log₂(-1)+log₂(-4)= log₂(-1)+log₂(1) + log₂(-1)+log₂(4) = log₂(-1×-1)+log₂(1)+log₂(4)=2

Hie to my teacher

I guess it is kind of randomly asking but does anybody know of a good website to watch new series online?

@Hassan Ruben I dunno I would suggest Flixportal. you can find it on google =) -kyng

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Pov that feelings when your just wanted to know more logarithmische in mathimatik but instead meeting Jesus the messiah!😂😂

Wow, that's not a good comment. 👎

And I literally did it🤦‍♀️🤦‍♀️

ah mannnn! yeah. you're totally right. thanks for pointing that out.

​@@ProfessorDaveExplains it's strange because when we compensate (-1) in the second expression after we condensed it, it's fine! but in the first expression (expanded) it's not valid as Teen pointed out.

Is this called as Critical Numbers?

The answer actually has two solutions because when you take the square root of both sides, you get answers. It should be -1 plus or minus e. ln stands for natural logarithm, which is the short for log base e. There are two that you can do that. The first method is to change this in exponential form, which (x+1)^2=e^2. Now take the square root of both sides with the plus or minus sign, which is x+1=plus or minus e. Therefore p, the answer is -1 pus or minus e. The second method is to use properties of logarithms. ln(x+1)^2=2ln(abs(x+1)). Notice that there is absolute value here because when you square the answer whether it’s positive or negative, the result will still yield the positive value. Divide both sides by 2. You have ln(abs(x+1))=1. Put this is exponential form, which is abs(x+1)=e. Remove the absolute by taking plus or minus sign on the right hand side, which is x+1=plus or minus e. Therefore, the answer is -1 plus or minus e.

no you can't, you can think of ln (log natural) as log e (x) so to cancel it out you need to use e e log e (x) = x

You can use either one as long as the base are the same. On your calculator, we only have common log (log base 10) and natural log (log base e). A logarithm without a subscript is referred to a common log (written as “log” on your calculator). log base e of x is the same as natural log (ln). Usually, we use ln only if the base is e and common log if the base is 10.

Same here, I’ve had a sub the past month and she’s really old and had to teach us this. Made no sense, and of course I’m only gonna try and actually learn it last minute now

it's just the definition of a log, the base of the log raised to the power of what's on the other side of the equals sign is equivalent to what the log is operating on.

If there’s no subscript written in the logarithmic form, it refers to the common log or log base 10.

@@ProfessorDaveExplains why is log 1 euqals to 0 but 10 in this equation?

​@@phewpheww4599 ​ there's a few parts to this answer. log(1) does = 0. Log(1) = x is asking you to find what number, if the base 10 was raised by this number, would be equal to 1? By the laws of exponents, any number raised to 0 will equal 1. Any base log will do the same. Log3(1) =0, log999(1) = 0. The answer to the third question, log(3x-2) = 1, has nothing to do with that rule. The 1 is already equal to the left side of the equation, the question is not asking you to find out what log(1) = x is. In the third question, we can express the 1 on the right as log(10), so it has the same base as the left side. Because log10(10) is equal to one, log(10) can be substituted for 1. To confirm that, solve log10(10) = x, which is asking by what number 10 needs to be raised by to equal 10. 10^1 = 10, so the answer is 1. Then we have log(3x-2) = log(10). As we have log on both sides, they can be cancelled out, leaving 3x-2 = 10. Balance by adding 2 to both sides. 3x = 12. Divide by 3. X = 4. I could be wrong but that's how I understand it.

Take ln of both sides, which is ln(5^(2x-3))=ln(11). Use the properties to bring the exponent down as a coefficient. (2x-3)ln(5)=ln(11). Divide both sides by ln(5) and add 3 then divide the whole answer by 2, which is (ln(11)/ln(5)+3)/2. If you want a more clear and exact answer, it would be (log_5(11)+3)/2. log_5(11) is log base 5 of 11. You can try this on your calculator if you want an approximate answer.

yes you're right👍

natural log! check out the three part series i did on logarithms that was released just before this one.

Professor Dave Explains Thanks

MILTIADIS POLITIS It's the natural logarithm. Basically, log to the base of e.

Well this isn't even calculus, it's algebra, so it sounds like you might be in a lot of trouble!

@@ProfessorDaveExplains omggg reallyyy aa but why is this whatmy calculus teacher lectured to us

I dunno what to tell you, this isn't calculus.

@@ProfessorDaveExplains i just got home from school and i just finished the exam! this video of yours actually helped tho even if the exam was calculus skskksks

Well, that's good news then! Make sure to keep watching through my mathematics playlist so that you're never in trouble again!

Just click here and pledge what you wish! www.patreon.com/professordaveexplains

Words cant explain how you have helped me thanks so much

The video is correct.

what?

ln is natural log, go to the tutorials earlier in the series explaining logs

Thank you didn't expected you

😮

​@@ProfessorDaveExplainssir which video please 🙏 tell

take log base 4!

Then go back earlier in the math series.

try setting it to 0.75 speed or even 0.5 speed