Finding y ′ of y = log 7 ⁡ e 8 x I know that...

2d3vljtq

2d3vljtq

Answered

2022-07-11

Finding y of y = log 7 e 8 x
I know that d d x ( e 8 x ) = 8 e 8 x , but I am confused on how to work the rest of the problem.
Is this correct:
log e x = ln x and that d d x ( ln x ) = y y = 1 x . (Can you show why log e x = ln x )
y = log b ( x ) b y = x ln ( b y ) = ln ( x ) y ln ( b ) = ln ( x ) y = ln ( x ) ln ( b )
d d x log b ( x ) = 1 x ln ( b )
log a ( x p ) = p log a x log 7 ( e 8 x ) = 8 x log 7 e
y = 8 e 8 x e 8 x ln ( 7 ) = 8 ln ( 7 )

Answer & Explanation

Jordan Mcpherson

Jordan Mcpherson

Expert

2022-07-12Added 16 answers

Yes, it's fine. An easier way to proceed is by noting that
y = x ( 8 log 7 e ) = x ( 8 ln e ln 7 ) = x 8 ln 7
by the change of base formula for logarithms.
Desirae Washington

Desirae Washington

Expert

2022-07-13Added 5 answers

Simply we have
y = log 7 e 8 x = ln e 8 x ln 7 = 8 x ln 7 y = 8 ln 7

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