SOLUTION: If log 2 base 10=0.431 and log 3 base 10=0.683, find the value of log 1/18 base 10

Question 849330: If log 2 base 10=0.431 and log 3 base 10=0.683, find the value of log 1/18 base 10
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
I'm not sure you are asking this problem right.

log10 (2) is not equal to .431

log10 (3) is not equal to .683

the base is not known.

If that's the case, your problem should be asking :

logb (2) = .431
logb (3) = .683
what is the value of logb (1/18)?

I was able to figure out that the base had to be 5, but I don't believe that is how they want you to solve this problem.

If I remember correctly, the way to solve this problem is as follows:

forget the base.
that is not known.
even though they tell you that if they show you the log without the base, you are to assume the base is 10, that is not what they mean in this case.

you have log(2) = .431 and log(3) = .683

you want to find log(1/18)

what you need to do is equate 1/18 in terms of 2 and 3.

1/18 is equivalent to 2/36.

2/36 is equivalent to 2 / (12 * 3)

2 / (12 * 3) is equivalent to 2 / (2 * 2 * 3 * 3)

now everything is in terms of 2 and 3 and you can apply the laws of logarithms to solve this problem.

your problem is to find log(1/18).

you solve your problem as follows:

log(1/18) = log(2 / (2*2*3*3)) = log[2 / (2^2 * 3^2)]

by the laws of logarithms:

log(a*b) = log(a) + log(b) *** multiplication law of logs
log(a/b) = log(a) - log(b) *** division law of logs
log(a^b) = b*log(a) *** power law of logs

apply the division law first to get:

log(1/18) = log(2) - [log(2^2 * 3^2)]

apply the multiplication law next to get:

log(1/18) = log(2) - [log(2^2) + log(3^2)]

simplify this by removing the outer parentheses (shown as []) to get:

log(1/18) = log(2) - log(2^2) - log(3^2)

apply the power law next to get:

log(1/18) = log(2) - 2 * log(2) - 2 * log(3)

since you already know what log(2) is equal to and what log(3) is equal to, you can replace log(2) and log(3) in this equation to get:

log(1/18) = .431 - 2 * .431 - 3 * .683

this can then be simplified to:

log(1/18) = -1.797

you did not need to know the base to solve this problem.

i did, however, solve for the base as follows:

logb(2) = .431 if and only if b^.431 = 2

if you raise both sides of this equation to the power of 1/.431, then this equation becomes:

(b^.431)^(1/.431) = 2^(1/.431)

simplify this equation to get:

b = 2^(1/.431)

use your calculator to find that b = 4.99396467

that's close enough to 5 for me to assume that 5 is probably the base.

assuming that 5 is the base, i get the following:

5^.431 = 2.00104139 which is very close to 2

5^.683 = 3.001902019 which is very close to 3

5^-1.797 = .0554563015 which is equal to .9982134271/18 which is very close to 1/18.

I was able to solve for the base but it was not necessary in order to solve the problem.

The way to solve the problem was shown above.

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