SOLUTION: If(y-1)log4 = ylog16, without using Mathematical table or calculator. Find the value of y. The bases are in 10.

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Question 1131605: If(y-1)log4 = ylog16, without using Mathematical table or calculator. Find the value of y.
The bases are in 10.
Found 3 solutions by MathLover1, Theo, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

%28y-1%29log%284%29+=+ylog%2816%29
log%284%5E%28y-1%29%29+=+log%2816%5Ey%29

log%284%5E%28y-1%29%29+=+log%28%284%5E2%29%5Ey%29

log%284%5E%28y-1%29%29+=+log%284%5E%282y%29%29........log same, then

4%5E%28y-1%29+=+4%5E%282y%29............bases same, then

y-1+=+2y

-1+=+2y-y

y=-1+







Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your equation is:

(y - 1) * log(4) = y * log(16).

y * log(16) is the same as y * log(4^2) which is the same as 2 * y * log(4).

your original equation becomes:

(y - 1) * log(4) = 2 * y * log(4)

divide both sides of this equation by log(4) and you get:

y - 1 = 2 * y

subtract y from both sides of this equation and you get:

-1 = 2 * y - y

simplify to get:

-1 = y

that's your solution.

confirm by replacing y in the original equation to get:

(y - 1) * log(4) = y * log(16) becomes (-1 - 1) * log(4) = -1 * log(16)

simplify to get -2 * log(4) = -1 * log(16).

use your calculator to get -1.204119983 = -1.204119983.

this confirms the solution is correct.

you did not use a calculator to solve this.
you only used a calculator to confirm the solution is correct.

the key to solving this is to realize that log(16) = log(4^2) and to realize that log (4^2) = 2 * log(4).

the fact that log(4^2) = 2 * log(4) is one of the properties of logs.

here's a reference on the properties of logs.

http://dl.uncw.edu/digilib/Mathematics/Algebra/mat111hb/EandL/logprop/logprop.html











Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

If(y-1)log4 = ylog16, without using Mathematical table or calculator. Find the value of y.
The bases are in 10.
matrix%281%2C3%2C+%28y+-+1%29+%2A+log+%28%284%29%29%2C+%22=%22%2C+y+%2A+log%2816%29%29

matrix%281%2C3%2C+log+%28%284%29%29%5E%28y+-+1%29%2C+%22=%22%2C+log+%28%2816%29%29%5Ey%29 ------ Applying matrix%281%2C3%2C+a+%2A+log%28%28b%29%29%2C+%22=%22%2C+log+%28%28b%29%29%5Ea%29

matrix%281%2C3%2C+log+%28%284%29%29%5E%28y+-+1%29%2C+%22=%22%2C+log+%28%284%5E2%29%29%5Ey%29 ------ Changing 16 to base 4, or 42



Since SAME-BASE logs are equal, we get: y - 1 = 2y

                                       - 1 = 2y - y

                                       highlight_green%28matrix%281%2C3%2C+-+1%2C+%22=%22%2C+y%29%29