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log2x+log4x=2

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4 Answers

This problem becomes simple if you understand the meaning of logarithms, and recall some simple rules.
 
Note that by the log conversion formula, Log4x = Log2x/Log24, so the problem becomes
 
log2x(1 + 1/Log24) = 2,  so (1.5)log2x = 2 or  log2x = 2/(1.5) = 4/3,  
 
Now remember, log2x = 4/3 means "the power that you raise 2 to in order to get x is 4/3", so x = the cube root of 24.  
 
x = cube root of 16.
 
 
 

Comments

Many students become flustered when dealing with logarithms because they forget the meaning of the word logarithm.  I ask all my students repeatedly
 
Q: "WHAT IS A LOGARITHM?"    (At first, you will get a blank stare followed by....I dunno ;-)
A: A logarithm is an EXPONENT   
 
Once students realize that the logarithm is simply the exponent of the base (b) the fear goes away
 
log10(100) is asking what the exponent of 10 is to produce 100....the exponent (logarithm) is '2'
 Log2 X + log 4 X = 2
 
 
  log4 X= m     X = 4^m = 2 ^(2m )
                      Log 4 X = 2 log2 X
 
 
    Log2 X +Log 4 X = Log2 X + 2 Log2 X = Log 2 X + Log 2 X^2 = Log 2 X^3 = Log 2 4
 
    Log (X .X^2)2 = Log2 4
 
     X^3 = 4
 
    X = 3√4

Comments

Parviz,
 
If you substitute X = 2(2/3) in the original equation, you get 1 not 2.  X is the cube root of 16, not the cube root of 4 as you stated.
Hi Aimee;
log2x+log4x=2
We need to work with the same base.
Let's convert one base.
I select the base of 4 because 22=4...
log4x
(log2x)/(log24)
(log2x)/2
(1/2)log2x
The equation is now...
log2x+(1/2)log2x=2
Let's combine like terms...
(3/2)log2x=2
Let's multiply both sides by 2/3...
log2x=4/3
x=24/3
The position of 3 in the denominator of the exponential means this is a cube root...
x=cube root(24)
x=cube root(16)