Evaluate Log Base 9 (9*sqrt 3)?

Question

Evaluate Please:

Log Base 9 (9*sqrt 3)

Here are my answer choices, please make sure to tell me how you arrived at this answer or it does me absolutely no good. Thanks!

a) -5/2
b) 5/2
c) -5
d) 4
e) 2

Hmm.. Well the last answer choice was "None of the above", but usually that's never the answer with this professor.. I guess I'll mark that (It's a practice test)! Thanks!

Answer 1

log_9 (9 * sqrt 3) = log_9 (9) + log_9 (sqrt 3)

Now sqrt(3) = sqrt(sqrt(9))
so
log_9 (9) + log_9 (sqrt 3) = log_9(9) + log_9(sqrt(sqrt(9))
= log_9(9) + log_9(9^(1/4))
= log_9 (9) + 1/4 log_9 (9)

Now log_9(9) = 1
so
log_9 (9 * sqrt 3) = 1 + 1/4 = 5/4

Hmm.. seems to be NONE of your options!

Answer 2

Ok, to solve this we need to work with the properties of logs:

log9(9) + log9(sqrt(3)) --- logb(mn) = logb(m)+logb(n)

Solve for each log separately.
log9(9)=1 --- logn(n) = 1

log9(3^(1/2))=(1/2)log9(3) --- logb(m^n)=nlogb(m)
x = log9(3) --- solve for log9(3) by making it equal to x
9^x=3 --- definition of log.
9^x=9^(1/2) --- 3=sqrt(9)=9^(1/2)
x=log9(3)=1/2 --- if m^a=m^b then a=b
So (1/2)log9(3)=(1/2)(1/2)=1/4

log9(9)+log9(sqrt3(3))=1+1/4=5/4
so
Log9(9sqrt(3))=5/4

I don't see that answer on their, perhaps you meant 5/4 instead of 5/2?

Answer 3

Log Base 9 (9*sqrt 3) = Log(9*sqrt(3)) /Log(9)=
{Log(9) + Log(sqrt(3))} /Log(9) =
1 + Log(sqrt(3))/Log(9) =
1 + (1/2Log3) / (2Log(3)) =
1 + 1/4 = 5/4

Answer 4

log9 (?x) - log3 (x^3) Write as an difficulty-free log expression with base e = ln(?x) / ln(9) -ln(x^3) / ln(3) when you consider that ?x= x^a million/2 and ln(9)= ln(3^2) you may now flow down the exponents by way of properties of logs = (a million/2)ln(x) / (2ln(3)) -ln(x^3) / (ln(3)) = (a million/4)ln(x) / ln(3) -ln(x^3) / ln(3) combine fractions = [ (a million/4)ln(x) -ln(x^3) ] / ln(3) by way of properties of logs = [(a million/4)ln(x) -3ln(x) ] / ln(3) = (-eleven/4)ln(x) / ln(3) OR (-2.seventy 5)log3 (x) that's base 3 i'm hoping this has been clean!

Answer 5

log(9)[9*sq rt 3]
log(9)9 + log(9)3^0.5
log(9)9 + 0.5log(9)3
log(9)9 = 1
0.5log(9)3 = 0.5 x 0.5 = 0.25
Hence log(9)9 + 0.5log(9)3 = 1 + 0.25 = 1.25 = 1 1/4

However, it may be easier to change to base 10. This is done by using
log(a)b = log(10)b / log10)a
So:-
log(9)[9*sqrt3) = log(10)[9*sqrt3] / log(10)9
= {log(10)9 + 0.5log(10)3} / log(10)9
In this base it is easier for your calculator to handle.
NB log(10)9 = 2 log(10)3
{2log(10)3 + 0.5log(10)3} / 2log(10)3
2.5 log(10)3 / 2 log(10)3 = 2.5 /2 (The 'log(10)3' cancels out.
2.5 /2 = 1.25 (As above).!!!!

Answer 6

log9(X) = ln(X) / ln(9) = ln(X) / 2ln(3) , for X > 0

ln(9 x V3) = ln(9) + ln(V3) = 2ln3 + 1/2.ln3 = 5/2.ln3

so:

Log Base 9 (9*sqrt 3) = log9(9 x V3) = (5/2.ln3) / (2.ln3) = 5/4.

None of your options is the right answer :-)