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Logarithm, please help?
Log of cubed root of x squared plus log of cubed root of x to the fourth equals log 2 to the -3.
Find x.
Thank you very much! But the answer in the back of the book is +/- square root of 2 over 4. Why is that?
1 Answer
- husoskiLv 79 years agoFavorite Answer
log[ (x^2)^(1/3) ] + log[ (x^4)^(1/3) ] = log[ 2^-3 ] ... changed radicals to fractional exponents
log[ x^(2/3) ] + log[ x^(4/3) ] = log[ 2^-3 ] .... simplify exponents on left using (a^b)^c = a^(bc) property
(2/3)log x + (4/3) log x = log[ 2^-3 ] ... apply log(a^b) = b log a property twice on left
2 log x = log [ 2^-3 ] ... combine like terms
log x = (1/2) log [ 2^-3 ] ... multiply both sides by 1/2
log x = log[ (2^-3)^(1/2) ] ... apply a*log b = log(b^a) property
x = (2^-3)^(1/2) ... if the logs are equal, so are the numbers
x = 2^(-3/2) .... and simply the exponents using (a^b)^c = a^(bc) property
That 1/sqrt(8) = sqrt(2)/sqrt(16) = sqrt(2)/4 ....after rationalizing the denominator
