Logarithmic functions? I am stuck on this question which as follows: log &#x2061;<!-- ⁡ --

Summer Bradford

Summer Bradford

Answered question

2022-06-21

Logarithmic functions?
log  ( x ) + log  ( x  3 ) = log  ( 10 x )
I have tried the following and not sure if I am doing it correctly...
1)
log  ( x ) + log  ( x  3 ) = log  ( 10 x )    log  ( ( x ) ( x  3 ) ) = log  ( 10 x )
   log  ( x 2  3 x ) = log  ( 10 x )    x 2  3 x = 10 x
x 2  13 x = 0    x = 0 , x = 13
2)
log  ( x ) + log  ( x  3 ) = log  ( 10 x )
log  ( x ) + log  ( x )  log  ( 3 ) = log  ( 10 x )
x + x  3 = 10 x    x =  3 / 8
Are these approaches on the right path?

Answer & Explanation

Jake Mcpherson

Jake Mcpherson

Beginner2022-06-22Added 23 answers

The initial strategy is the best one. Not the second is: log  ( x  3 )  log  x  log  3
The following stage is to confirm which solutions are necessary to eliminate any unnecessary ones ( x = 0 and x = 13) satisfy the original equation.

fabios3

fabios3

Beginner2022-06-23Added 10 answers

I am not sure if l o g ( z ) means l o g 10 ( z ) or l o g e ( z ) i.e l n ( z ) so I will go with convention and assume l o g ( z )    l o g 10 ( z )
Here goes:
so x  { 0 , 13 }
But for x = 0 the original equation becomes: l o g ( 0 ) + l o g (  3 ) = l o g ( 0 ) his is a problem because l o g ( z ) is defined only for 0 < z if z is real.
Complex number logarithms quickly get disorganized. The only response is left when the log function arguments are limited to positive reals: x = 13

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