Question

How do you solve `\log _ { 27} ( 2x + 4) + \log _ { 81} ( 6x + 12) = \frac { 5} { 6}`?

Answer 1

`x=-1/2`

Explanation:

Using `logab=logb/loga`, we can simplify given equation and solve it as follows:

`log_27(2x+4)+log_81(6x+12)=5/6`

i.e. `log(2x+4)/log27+log(6x+12)/log81=5/6`

or `log(2x+4)/(3log3)+log(6x+12)/(4log3)=5/6`

or `log(2x+4)/3+log(6x+12)/4=5/6log3`

or `4log(2x+4)+3log(6x+12)=10log3`

or `log[(2x+4)^4(6x+12)^3]=log3^10`

or `(2x+4)^4(6x+12)^3=3^10`

or `2^4(x+2)^4*6^3(x+2)^3=3^10`

or `(x+2)^7=3^10/(2^4*6^3)`

or `(x+2)^7=3^10/(2^4*2^3*3^3)`

or `(x+2)^7=3^7/2^7`

or `x+2=3/2`

or `x=3/2-2=-1/2`