Question 283809: 2log(7,3) + 3log(7,2) = log(7, x)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 2*log(7,3) + 3*log(7,2) = log(7,x)
We can rewrite this as:
log(7,x) = 2*log(7,3) + 3*log(7,2)
The rules of logarithms that apply here will be:
log(a^n) = n*log(a)
log(a*b) = log(a) + log(b)
First thing we notice is that 2*log(7,3) would be the same as log(7,3^2) which would be the same as log(7,9).
Second thing we notice is that 3*log(7,2) would be the same as log(7,2^3) which would be the same as log(7,8).
Our equation becomes:
log(7,x) = log(7,9) + log(7,8)
Third thing we notice is that log(7,9) + log(7,8) would be the same as log(7,9*8) which would be the same as log(7,72)
Our equation becomes:
log(7,x) = log(7,72)
This can only be true if x = 72.
Our original equation is:
log(7,x) = 2*log(7,3) + 3*log(7,2)
We replace x with 72 and we get:
log(7,72) = 2*log(7,3) + 3*log(7,2)
We can convert this equation to the base of 10 so we can solve using our calculator.
The conversion formula in general is:
log(b,x) = log(c,x)/log(c,b)
This reads as log of x to the base b = log of x to the base c divided by the log of b to the base c.
The conversion formula in our case is:
log(7,x) = log(10,x) / log(10,7)
Our equation becomes:
log(10,72/log(10,7) = 2*log(10,3)/log(10,7) + 3*log(10,2)/log(10,7)
If we multiply both sides of this equation by log(10,7), then we get:
log(10,72) = 2*log(10,3) + 3*log(10,2) which becomes:
1.857332496 = 2*.477121255 + 3*.301029996 which becomes:
1.857332496 = 1.857332496 which is true confirming the equations are equivalent.
Your answer is that x = 72.
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