Simple Log Question Fairly new to logs, stuck here: The equation y = 4 <mrow

Waylon Ruiz

Waylon Ruiz

Answered question

2022-05-23

Simple Log Question
Fairly new to logs, stuck here:
The equation y = 4 3 x can be written in terms of x:
y = 4 3 x
log ( y ) = log ( 4 3 x )
0 = 3 x log ( 4 ) log ( y )
0 = 3 x log ( 4 y )
At this point I am totally stumped. I must be missing something obvious here.

Answer & Explanation

Cordell Crosby

Cordell Crosby

Beginner2022-05-24Added 11 answers

The quotient rule of logarithms says that for any positive b 1 and any positive M , N we have
log b M log b N = log b M N .
Note that this says nothing about differences of multiples of logarithms. The 3 x is going to prevent you from bringing them together in that way. Instead, from
3 x log 4 = log y ,
we can simply divide through by 3 log 4 to get
x = log y 3 log 4 .
Here is another way to approach it. Given any positive b 1 and any positive t , we can think of log b t as "the exponent on the base b that yields t." That is, log b t is the unique number s such that t = b s . That means (for instance) that we can immediately rewrite y = 4 3 x as 3 x = log 4 y , so that
x = 1 3 log 4 y .
From there, we can use the change-of-base formula to rewrite it in terms of common logarithms, if we like, and get exactly the same result as above.
Alani Conner

Alani Conner

Beginner2022-05-25Added 3 answers

The final step is not correct:
3 x log ( 4 y ) = 3 x ( log 4 log y ) = 3 x log 4 3 x log y

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