SOLUTION: What is the value of x? https://i.ibb.co/5FdV518/2.jpg (Sorry I don't know how to type this up. I looked at the guide but still can't get it right.)

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: What is the value of x? https://i.ibb.co/5FdV518/2.jpg (Sorry I don't know how to type this up. I looked at the guide but still can't get it right.)      Log On


   

Question 1204568: What is the value of x?
https://i.ibb.co/5FdV518/2.jpg
(Sorry I don't know how to type this up. I looked at the guide but still can't get it right.)
Found 3 solutions by josgarithmetic, math_tutor2020, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
log(9,3*sqrt(3))=x

log%289%2C3%2Asqrt%283%29%29=x

--

9%5Ex=3sqrt%283%29
%283%5E2%29%5Ex=3%5E1%2A3%5E%281%2F2%29
3%5E%282x%29=3%5E%283%2F2%29
same base both sides so the exponents are equal.
2x=3%2F2
highlight%28x=3%2F4%29

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

To type log%289%2C%28x%29%29 on a keyboard, you could write log(9,x)

So something like log%289%2C%283%2Asqrt%283%29%29%29 would be log(9, 3*sqrt(3) )


Now onto the problem at hand.
log%289%2C%283%2Asqrt%283%29%29%29+=+x

9%5Ex+=+3%2Asqrt%283%29 Convert the equation to exponential form

%283%5E2%29%5Ex+=+3%5E1%2A%283%29%5E%281%2F2%29%5E%22%22 Rewrite 9 as 3^2. Rewrite the square root as an exponent of 1/2.

3%5E%282x%29+=+3%5E1%2A%283%29%5E%281%2F2%29%5E%22%22 Use the rule that (a^b)^c = a^(b*c)

3%5E%282x%29+=+3%5E%281%2B1%2F2%29%5E%22%22 Use the rule that a^b*a^c = a^(b+c)

3%5E%282x%29+=+3%5E%283%2F2%29%5E%22%22

2x+=+3%2F2 Since the bases are both 3, the exponents must be equal.

x+=+%283%2F2%29%2A%281%2F2%29

x+=+3%2F4

-----------------------------

Another approach would be to use the change of base formula.
log%289%2C%283%2Asqrt%283%29%29%29+=+x

x+=+log%289%2C%283%2Asqrt%283%29%29%29

x+=+log%28%283%2Asqrt%283%29%29%29%2F%28log%28%289%29%29%29 Change of base formula is applied here. From this point onward, the logs are base 10.

x+=+log%28%283%5E1%2A%283%29%5E%281%2F2%29%29%29%2F%28log%28%283%5E2%29%29%29

x+=+log%28%283%5E%281%2B1%2F2%29%5E%22%22%29%29%2F%28log%28%283%5E2%29%29%29

x+=+log%28%283%5E%283%2F2%29%5E%22%22%29%29%2F%28log%28%283%5E2%29%29%29

x+=+%28%283%2F2%29%2Alog%28%283%29%29%29%2F%282%2Alog%28%283%29%29%29 Use the log rule log(A^B) = B*log(A)



x+=+%28%283%2F2%29%2Across%28log%28%283%29%29%29%29%2F%282%2Across%28log%28%283%29%29%29%29

x+=+%283%2F2%29%2F%282%29

x+=+matrix%281%2C3%2C3%2F2%2C%22divide%22%2C2%29

x+=+%283%2F2%29%2A%281%2F2%29

x+=+3%2F4
Your steps do not have to be as verbose as shown above.

As quick confirmation, use a calculator to find that:
log(3*sqrt(3))/log(9) = 0.75

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


A different approach -- easier, in my opinion...

Use log%28a%2C%28b%29%29=log%28%28b%29%29%2Flog%28%28a%29%29

For this problem, we have

log%289%2C%283sqrt%283%29%29%29=log%28%283sqrt%283%29%29%29%2Flog%28%289%29%29

In this expression we can use any base we want. Since all the numbers are powers of 3, use log base 3.

log%283%2C%283sqrt%283%29%29%29%2Flog%283%2C%289%29%29=1.5%2F2=3%2F4

ANSWER: 3/4