Solve for x log2 (x) +log3 (x)=1.
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We have to solve for x given that log(2) x + log(3) x = 1
log(2) x + log(3) x = 1
take the log to the base 10
log x/log 2 +...
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We'll change the bases of logarithms into the common base 10.
log2 (x) = log x/log 2
log3 (x) = log x/log 3
We'll re-write the equation:
log x/log 2 + log x/log 3 = 1
We'll calculate the LCD = (log 2)*(log 3)
We'll multiply all over by (log 2)*(log 3):
(log 2)*(log 3)*(log x)/log 2 + (log 2)*(log 3)*(log x)/log 3 = (log 2)*(log 3)
(log 3)*(log x) + (log 2)*(log x) = (log 2)*(log 3)
We'll factorize by log x:
(log x)*[(log 3) + (log 2)] = (log 2)*(log 3)
We'll apply the product property of logarithms:
log x = (log 2)*(log 3)/log(2*3)
log x = (log 2)*(log 3)/log 6
Since the base is 10, we'll take antilog and we'll get:
x = 10^[(log 2)*(log 3)/log 6]
The solution of the given equation is x = 10^[(log 2)*(log 3)/log 6].
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