SOLUTION: log4 (x-3)-log16 (x-3)=1

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Question 991960: log4 (x-3)-log16 (x-3)=1
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
one way to do this is to convert everything to base of 10.

log4(x-3) = log(x-3)/log(4)

log16(x-3) = log(x-3)/log(16)

your original equation of log4(x-3) - log16(x-3) = 1 becomes:

log(x-3)/log(4) - log(x-3)/log(16) = 1

multiply both sides of the equation by log(4) * log(16) and you get:

log(x-3)*log(16) - log(x-3) * log(4) = 1 * log(4) * log(16)

factor out log(x-3) on the left side of the equation to get:

log(x-3) * (log(16) - log(4)) = 1 * log(4) * log(16)

divide both sides of the equation by (log(16) - log(4)) to get:

log(x-3) = (1 * log(4) * log(16)) / (log(16) - log(4))

solve the logarithms on the right side of the equation to get:

log(x-3) = 1.204119983

this is true if and only if 10^1.204119983 = x-3

solve for x in this equation and you get:

x = 10^1.204119983 + 3

this results in x = 19.

replace x in the original equation with 19 and you will see that the original equation is true which confirms the solution is correct.

another way to do this is to convert everything to base of 4.

log16(x-3) is equal to log4(x-3) / log4(16)

equation becomes:

log4(x-3) - log4(x-3) / log4(16) = 1

now log4(16) is equal to y if and only if 4^y = 16.
this makes y = 2 because 4^2 = 16, so you get:
log4(16) = 2

this makes the equation of log4(x-3) - log4(x-3) / log4(16) = 1 become:

log4(x-3) - log4(x-3) / 2 = 1

multiply both sides of this equation by 2 to get:

2*log4(x-3) - log4(x-3) = 2

since 2*log4(x-3) is the same as log4((x-3)^2), the equation becomes:

log4((x-3)^2) - log4(x-3) = 2

since log4((x-3)^2) - log4(x-3) is the same as log4((x-3)^2/(x-3)), the equation becomes:

log4((x-3)^2/(x-3)) = 2

since (x-3)^2/(x-3) is the same as (x-3), the equation becomes:

log4(x-3) = 2

this is true if and only if 4^2 = x-3 which becomes:

16 = x - 3

solve for x to get x = 19.

either way you get x = 19, so that's your solution.

you had to convert your logs to a common base either way.











Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
log%284%2C%28x-3%29%29 - log%2816%2C%28x-3%29%29 = 1.

Notice that log%2816%2C%28x-3%29%29 = 1%2F2log%284%2C%28x-3%29%29,     since     16 = 4%5E2.

So your equation takes the form

log%284%2C%28x-3%29%29 - 1%2F2log%284%2C%28x-3%29%29 = 1,     or     1%2F2log%284%2C%28x-3%29%29 = 1,     or     log%284%2C%28x-3%29%29 = 2.

The last equation means that

log%282%2C%28x-3%29%29 = 4,     since     4 = 2%5E2.

Hence,  x - 3 = 2%5E4 = 16  and then  x = 19.

Answer.  x = 19.