Question

Explain the process the properties you have to use to solve the logarithmic equation: log3x + log34 - 2 log33 = 2

Answer 1

First, we are going to isolate
`log3x` in the left side of the equation:

`log3x+log34-2log33=2`

`log3x=2+2log33-log34`

Next, we are going to use log of a power rule:
`nlogx=logx^(n)`

`log3x=2+2log33-log34`

`log3x=2+log33^(2)-log34`

`log3x=2+log1089-log34`

Next, we are going to use the log of a quotient rule:
`loba-logb=log (a)/(b)`

`log3x=2+log1089-log34`

`log3x=2+log (1089)/(34)`

Next, we are going to use the rule:
`a=log_(b)b^(a)`

`log3x=2+log (1089)/(34)`

`log3x=log10^{(2+log (1098)/(34))`

`log3x=log[(10^(2))(10^{log (1089)/(34) })]`

`log3x=log[(100)( (1089)/(34) )]`

`log3x=log (54450)/(17)`

And last but not least, we are going to use the same base log rule:
`loga=logb` →
`a=b`

`log3x=log (54450)/(17)`

`3x= (54450)/(17)`

`x= (54450)/((3)(17))`

`x= (18150)/(17)`

We can conclude that the solution of our logarithmic equation is
`x= (18150)/(17)`.