Question

Which of the following shows the extraneous solution to the logarithmic equation below?
log3(18x^3)-log3(2x)=log3(144)

x=-16
x=-8
x=-4
x=-2
I know C is not the extraneous solution, because that is the valid solution

Answer 1

well doesn't extraneous mean that it maybe a solution but when you check it, the solution doesn't exist...

so substitute you solution into the original equation....

you need to remember that you can't take the log of a negative number....

Answer 2

so substitute your choice for a valid solution into the original equation and see if it works..

Answer 3

Can you maybe work out the full problem for me? Because I'm not really sure of the other solution options other than 4 and -4.

Answer 4

ok... x = -4 as a solution

\[\log_{3}[18 \times (-4)^3] - \log_{3}[2 \times (-4)] = \]

what is the value of the numbers inside the square brackets..?

Answer 5

one of the fundemental rules for working with logs is that you can't take the log of a negative number.....

so think about that as you calculate the values.. of

\[18 \times (-4)^3 ~~and~~ 2 \times (-4) \]

Answer 6

-1152 and -8. So would the answer to the problem be -8?

Answer 7

no... the fact is now you are looking at

\[\log_{3}(-1152) - \log_{3}(-8)\]

and as I said previously you can't take the log of a negative number...

so while x = -4 is a solution to the equation when simplified, the fact that when you substitute it into the original equation you end up with

\[\log_{3}(-1152) - \log_{3}(-8) = \log_{3}(144)\]

it doesn't work...

as an example calculate ln(-5) what do you get...?

Answer 8

when you put the positive solution in x = 4 you get

\[\log_{3}(1152) - \log_{3}(8) = \log_{3}(144)\]

which is correct...

Answer 9

So what would the answer be then, since 8 or -8 does not work?

Answer 10

the extraneous solution is x = -4 since you can't take the log of a negative number..... as I've said previously.

x = 4 is a valid solution....