I am trying to proof the following expression (without a calculator of course). log_(12)18xxlog_(24)54+5(log_(12)18−log(24)54)=1 I know this isn't a difficult task but it's just killing me. I have tried many things, among which was base transformation to 12 and expressing every logarithm in terms of log_(12)3 and log_(12)2 but every time I try to do it, I mess up something. I don't know if my concentration is terrible or I'm doing something wrong.

Cyrus Travis 2022-10-22 Answered
Proof the expession log 12 18 × log 24 54 + 5 ( log 12 18 log 24 54 ) = 1
I am trying to proof the following expression (without a calculator of course).
log 12 18 × log 24 54 + 5 ( log 12 18 log 24 54 ) = 1
I know this isn't a difficult task but it's just killing me. I have tried many things, among which was base transformation to 12 and expressing every logarithm in terms of log 12 3 and log 12 2 but every time I try to do it, I mess up something. I don't know if my concentration is terrible or I'm doing something wrong.
Thanks ;) ( if there are more levels to this task, I'd like a hint, not a complete solution)
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

honotMornne
Answered 2022-10-23 Author has 12 answers
Let log 12 18 = a , log 24 54 = b. So you want to prove that a b + 5 ( a b ) = 1
Then you get 12 a = 18 and 24 b = 54. Now factor everything in powers of 2 and 3 to get
2 2 a 1 = 3 2 a 2 3 b 1 = 3 3 b .
From this taking log base 2 you will get:
2 a 1 = ( 2 a ) log 2 3 3 b 1 = ( 3 b ) log 2 3.
Furthermore you get
2 a 1 2 a = 3 b 1 3 b .
Now simplify this and see you will get the expression written on the first line.
Did you like this example?
Subscribe for all access
Sonia Elliott
Answered 2022-10-24 Author has 4 answers
The follow-your-nose approach would be, in my opinion,
log 12 18 log 24 54 + 5 ( log 12 18 log 24 54 ) = log 18 log 12 log 54 log 24 + 5 ( log 18 log 12 log 54 log 24 ) = log 18 log 12 log 54 log 24 + 5 ( log 18 log 24 log 54 log 12 log 12 log 24 ) = log 18 log 54 + 5 log 18 log 24 5 log 54 log 12 log 12 log 24 .
To save space, write t = log 2 and h = log 3. Then log 18 = log 2 + 2 log 3 = t + 2 h, etc., and we get
log 18 log 54 + 5 log 18 log 24 5 log 54 log 12 log 12 log 24 = ( t + 2 h ) ( t + 3 h ) + 5 ( t + 2 h ) ( 3 t + h ) 5 ( t + 3 h ) ( 2 t + h ) ( 2 t + h ) ( 3 t + h ) = t 2 + 5 t h + 6 h 2 + 15 t 2 + 35 t h + 10 h 2 ( 10 t 2 + 35 t h + 15 h 2 ) 6 t 2 + 5 t h + h 2 = 1.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-05-13
How do you solve this logarithm?
Solve for x in the following:
x = 9 log 3 ( 2 )
The answer is 4, but why?
asked 2021-12-31
Show that log(x+1)log(x)<1x for x>0
asked 2021-11-06
Given that loga(5)0.65 and loga(3)0.44, evaluate each of the following. Hint: use the properties of logarithms to rewrite the given logarithm in terms of the logarithms of 5 and 3
a)loga(0.6)
b)loga(3)
c)loga(15)
d)loga(25)
e)loga(75)
f)loga(1.8)
asked 2022-11-18
How many numbers less than x have a prime factor that is not 2 or 3
I am trying to figure out the number of integers greater than 1 and less than or equal to x that have a prime factor other than 2 or 3. For example, there are only two such integer less than or equal to 7.
It is straight forward to determine how many many integers less than or equal to x have a prime factor other than 2:
x log 2 x
Or to make the same determination about 3:
x log 3 x
What is the method or formula for figuring out how many integers less than or equal to x have a prime factor other than 2 or 3?
I know that it is less than:
x log 2 x log 3 x
and greater than:
x log 2 x log 3 x x 6
Thanks
asked 2022-10-14
Help with Evaluating a Logarithm
A precalculus text asks us to evaluate log 8 2 256 3 32 6
I do the following: log 8 2 ( 2 2 ) 3 2 2 3 2 3 2 2 6
log 8 2 2 2 2 2 3 2 2 2 6
log 8 2 2 2 2 3 2 3
and then I'm stumped.
Hints?
asked 2021-10-13
Write the following expression as a sum and/or difference of logarithms. Express powers as factors.
ln(e48)
asked 2022-03-28
Consider the exponential equation. 9x=733
a. Find x in exact form: x=b. Approximate x, correct to at least 3 decimal places: x=
b. Approximate x, correct to at least 3 decimal places: x=