SOLUTION: How many real solutions does this equation have? {{{log(2014, (x)) - log(x, (2014))= log(root(2014, 2014), (root(2014, x)))}}}

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: How many real solutions does this equation have? {{{log(2014, (x)) - log(x, (2014))= log(root(2014, 2014), (root(2014, x)))}}}      Log On


   

Question 1053688: How many real solutions does this equation have?
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52908) About Me  (Show Source):
You can put this solution on YOUR website!
.
How many real solutions does this equation have?

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Answer. There are no real solutions.

Solution 

1.  In the left side  log%28x%2C+2014%29 = 1%2Flog%282014%2Cx%29  (use the formula of the base change for logarithm; see the reference below).


2.  In the right side

    log%28root%282014%2C+2014%29%2C+root%282014%2C+x%29%29 = 2014%2Alog%282014%2C+root%282014%2C+x%29%29 = %282014%2F2014%29%2Alog%282014%2Cx%29 = log%282014%2Cx%29.


3.  Therefore, the given equation takes the form 

    log%282014%2C+x%29 - 1%2Flog%282014%2Cx%29 = log%282014%2Cx%29.

    It implies that  -1%2Flog%282014%2Cx%29 = 0, which has no real solutions.

On properties of logarithms, see the lessons
    - WHAT IS the logarithm
    - Properties of the logarithm
    - Change of Base Formula for logarithms
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(20065) About Me  (Show Source):
You can put this solution on YOUR website!


How many real solutions does this equation have?


We will assume that x is positive and %22%22%3C%3E%22%221.

Let a = 2014



Use identity log%28B%2C%28C%29%29=1%2Flog%28C%2C%28B%29%29 and matrix%282%2C1%2C%22%22%2Croot%28N%2CM%29=M%5E%281%2FN%29%29



Use a rule of logarithms: log%28B%2C%28C%5ED%29%29=D%2Alog%28B%2C%28C%29%29



Use identity log%28B%2C%28C%29%29=1%2Flog%28C%2C%28B%29%29



Use identity: matrix%282%2C1%2C%22%22%2Croot%28N%2CM%29=M%5E%281%2FN%29%29



Use a rule of logarithms: log%28B%2C%28C%5ED%29%29=D%2Alog%28B%2C%28C%29%29





1%2Flog%28x%2C%28a%29%29+-+log%28x%2C+%28a%29%29=++1%2Flog%28x%2C%28a%29%29

Let y = logx(a)

1%2Fy+-+y=++1%2Fy

1+-+y%5E2+=+1+

-y%5E2=0

y%5E2=0

y=0

logx(a)=0

x%5E0=a

x%5E0=2014

1=2004

That's false so there are no solutions.

Edwin