Question

Which of the following expressions does NOT have the same derivative as y=log_3 x

Answer 1

Which of the following expressions does not have the same derivative as
\[y=\log_{3} x\]
a)\[\frac{ 1 }{ 2 }\log_{6}x \]
b)\[\log_{9}x^2, x >0\]

My guess is B, but i dont see how a has the same derivative as \[y=\log_{3} x\]

Answer 2

Do you know what the derivative of \(\log_3x\) is?

Answer 3

derivative of lnx/ln3

Answer 4

which should be 3/x i think

Answer 5

Hmm, not quite.
\[\frac{d}{dx}\log_3x=\frac{d}{dx}\frac{\ln x}{\ln 3}=\frac{1}{x\ln3}\]

Answer 6

o wait, i get it, b is 6/x then u divide that by 2, and u get 3/x?

Answer 7

Your derivatives aren't right, but your reasoning is.

Answer 8

>.> forgot that ln3 is a constant multiple >.>

Answer 9

the derivative of a should be 1/2xln6
and derivative of b should be 2/xln9 correct?

Answer 10

\[y=\frac{1}{2}\log_6x~~\implies~~y'=\frac{1}{2}\frac{\ln x}{\ln6}=\frac{\ln x}{\ln36}\]
Hmm looks like (a) isn't right.

Answer 11

in my opinion, neither have the same derivative as log_3 of x

how'd u get ln36 on the bottom btw?

Answer 12

It's a property of logarithms. \(a\log_bx=\log_bx^a\).

Answer 13

but how does that get u ln 36?

Answer 14

In the denominator,
\[2\ln6=\ln6^2=\ln36\]

Answer 15

i get it, and just to confirm, neither of these have the same derivative as log_3 x?

Answer 16

\[y=\log_9x^2=\frac{\ln x^2}{\ln9}=\frac{2\ln x}{\ln 9}~~\implies~~y'=\frac{2}{x\ln9}=\frac{1}{\frac{1}{2}x\ln 9}=\frac{1}{x\ln9^{1/2}}\]
Nope, it's (b).

Answer 17

how do u get the 1/2 on the bottom?

Answer 18

by multiplying the top and bottom by 1/2?