log help！ (1 Viewer)

[darkphoenix]

Hey, i was doing log question and I cannot get the meaning of e^x
The question is to differentiate log10x and the answer is (log10e)/x
can some one explain why the e come out from no where. Thanks heaps

 

[4025808]

Hey, i was doing log question and I cannot get the meaning of e^x
The question is to differentiate log10x and the answer is (log10e)/x
can some one explain why the e come out from no where. Thanks heaps

log10(x) = loge(x)/loge(10) ------> change of base rule

now that 1/loge(10) is a constant, we leave it unchanged. then we differentiate loge(x) which gives 1/x

1/x * 1/loge(10) = 1/[xloge(10)]

btw the answer you provided is wrong, to confirm, go here -> http://www.wolframalpha.com/input/?i=differentiate+log+base+10+x

 

[darkphoenix]

OH, ok. Thanks, actually the full question(excel)is to differentiate with respect to x: (log10x)^3

and the solution is 1/x(3log10e)(log10x)^2

 

[SpiralFlex]

OH, ok. Thanks, actually the full question(excel)is to differentiate with respect to x: (log10x)^3

and the solution is 1/x(3log10e)(log10x)^2

In this case you cannot fully utilise your logarithmic laws. Or can we? Change of base.

First of all I will clearly use the change of base law.



Now we can finally derive!

 

[darkphoenix]

Sorry I just started to do log in this holiday, I am confused at change of base formula. log a(b)= logc(b) /logc(a) can c be any thing? or just e

 

[math man]

In this case you cannot fully utilise your logarithmic laws. Or can we? Change of base.

First of all I will clearly use the change of base law.



Now we can finally derive!

answer should be

 

[darkphoenix]

So, confused, which one is right? all different answers

 

[math man]

i just saw the asnwer..to change my answer to the right answer you realise:



this simplys to:



and subbing this into my answer gives the desired result:

 

[math man]

fine full solution is:

using change of base rule.

using power rule.

Edit: full worked solution to obtain 1/x(3log10e)(log10x)^2 is:

from



we realise that:



and that

by applying change of base rule for 10 now

subbing these into my first derivate becomes:




and this becomes:



which is the solution you gave us

 

[nightweaver066]

 

[D94]

answer should be

This is one version which is correct.

Edit: and above is another correct version.

 

[math man]

Sorry I just started to do log in this holiday, I am confused at change of base formula. log a(b)= logc(b) /logc(a) can c be any thing? or just e

and we can use the change of base for anything, but we only know how to differentiate base e so thats why we change it to e in these questions

 

[SpiralFlex]

Oops left out a part.

 

[darkphoenix]

and we can use the change of base for anything, but we only know how to differentiate base e so thats why we change it to e in these questions

finally I understand, thank you and the others