Question 380916: write the expression (log base of b)(2y+5)-4(log base of b)(y+3) as a single
logarithm
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! that expression would be shown as:
log(b,(2y+5)) - 4*log(b,(y+3))
in general x * log(y) = log(y^x)
your expression becomes:
log(b,(2y+5)) - log(b,((y+3)^4))
in general log(x) - log(y) = log(x/y)
your expression becomes:
log(b,((2y+5)/(y+3)^4))
to show you how this works, we will let b = 10 because your calculator can do logs to the base of 10 (usually called the LOG function).
we will let y = 5 (chosen at random small enough to calculate easily).
your original expression becomes:
log(2y+5) - 4*log(y+3)
the base of 10 is implied.
substituting 5 for y, we get:
log(2*5+5) - 4*log(5+3) which becomes:
log(15) - 4*log(8) which becomes -2.436268689
looking at our final expression of:
log(b,(2y+5)/(y+3)^4), we get:
log((2y+5)/(y+3)^4).
the base of 10 is implied.
substituting 5 for y, we get:
log(15/8^4) which becomes log(15/4096).
using our calculator, we get log(15/4096) = log(.003662109) which equals -2.436268689
we get the same answer either way, so the translation is good, and the answer to your question is:
log(b,2y+5) - 4*log(b,y+3) = log(b,(2y+5)/(y+3)^4) which looks like:
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