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

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

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Answer. There are no real solutions.

Solution 

1.  In the left side   =   (use the formula of the base change for logarithm; see the reference below).


2.  In the right side

     =  =  = .


3.  Therefore, the given equation takes the form 

     -  = .

    It implies that  - = 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)   (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 1.

Let a = 2014



Use identity  and 



Use a rule of logarithms: 



Use identity 



Use identity: 



Use a rule of logarithms: 







Let y = logx(a)











logx(a)=0







That's false so there are no solutions.

Edwin
RELATED QUESTIONS

Compare 2014^2015 and... (answered by josgarithmetic)
Find the maximum integral value of k such that 2014^k divides (1! x 2! x 3!x ... x... (answered by greenestamps)
How many ordered pairs of positive integers (a,b) satisfy the following equation? 2a + 3b (answered by LinnW)
find the value of 6x^3 + x^2 - 7x + 2014 if 2x^2 + x... (answered by Edwin McCravy)
how many real solutions does the following equation have?... (answered by Theo)
In 2014, quarterly sales for full-service restaurants and shoe stores in the United... (answered by ikleyn)
what is 15 multiplied by... (answered by jim_thompson5910)
If x = 2015, y = 2014 and z = 2013, then the value of x² + y² +z² – xy – yz – zx... (answered by ikleyn)
(sigma i = 1 to 2014 i^2 ) / (sigma i = 1 to 2014 i )... (answered by Edwin McCravy)