SOLUTION: Please help. Show all your work. Round your answers to four decimal places log2 (c-12) +log2 (2c)=7

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Question 1077495: Please help. Show all your work. Round your answers to four decimal places
log2 (c-12) +log2 (2c)=7
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52817) About Me  (Show Source):
You can put this solution on YOUR website!
.
log2 (c-12) +log2 (2c)=7  --->  

(c-12)*(2c) = 2%5E7   --->

(c-12)*c = 2%5E6  --->

c%5E2+-+12c+-+64 = 0,

(C-16)*(c+4) = 0  --->

c = 16  and/or  c = -4 --->

c = 16 is the unique solution which survives under the logarithm.


Answer.  c = 16.

Solved.


On logarithms, see the lessons
    - WHAT IS the logarithm
    - Properties of the logarithm
    - Change of Base Formula for logarithms
    - Solving logarithmic equations
    - Using logarithms to solve real world problems
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Logarithms".



Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
log%282%2C%28c-12%29%29+%2Blog%282%2C%282c%29%29%22%22=%22%227

Use the rule of logarithm that says that the sum of
two logs equals the log of the product of what the
two logs are taken of.  That rule is usually written

    log%28b%2C%28X%29%29%2Blog%28b%2C%28Y%29%29=log%28b%28XY%29%29.

Therefore

log%282%2C%28c-12%29%29+%2Blog%282%2C%282c%29%29%22%22=%22%227

becomes

log%282%2C%28%28c-12%29%5E%22%22%282c%29%5E%22%22%29%29%29%22%22=%22%227 

Use the rule of logarithms that says that a logarithm equals
the exponent to which its base must be raised in order
to get the number that the log is being taken of.  That is 
usually written:

   log%28b%2C%28X%29%29%22%22=%22%22Y is equivalent to X%22%22=%22%22b%5EY

Therefore

log%282%2C%28%28c-12%29%5E%22%22%282c%29%5E%22%22%29%29%29%22%22=%22%227  becomes

%28c-12%29%282c%29%22%22=%22%222%5E7


2c%28c-12%29%22%22=%22%22128

2c%5E2-24c%22%22=%22%22128

Divide through by 2

c%5E2-12c%22%22=%22%2264

Get 0 on the right by subtracting 64 from both sides:

c%5E2-12c-64%22%22=%22%220

%28c-16%29%28c%2B4%29%22%22=%22%220

c-16 = 0;   c+4 = 0
   c = 16     c = -4

We must ignore the answer c=-4 as extraneous
because in the original equation:

log%282%2C%28c-12%29%29+%2Blog%282%2C%282c%29%29%22%22=%22%227

if we substitute -4 we will have the logarithm of
a negative number which is not defined in ordinary
mathematics.  However the answer 16 is valid.

Edwin