Page 85 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I equAtions

1
1
If a> 0 but a =1, then x> 0,x =1,x = ,x = , and log a = 1 , log a = log a = 1 , log 2 a = log a = 1
a
a
a a 2 x log x ax log a+log x 1+log x a x 2 log a+log x 2+log x
a
a
a
a
a
a
a
log a
log a
1
1
1
log a = log x , log a = log a+log x = 1+log x , log 2 a = 2 log a+log x = 2+log x . The original equation is equivalent to
a
a
x
ax
a x
a
a
a
a
a
a
a
2
2 + 1 + 3 =0 . Let t = log x , then 2 1 3 6t +11t+4 2
2
=0 ⇒ 6t + 11t +4 = 0 ⇒ (3t + 4)(2t + 1) = 0 ⇒
2
=0 ⇒ 6t +11t+4
log x 1+log x 2+log x a 2 t + + 1 1+t + + 3 2+t =0 ⇒ t(1+t)(2+t) =0 ⇒ 6t + 11t +4 = 0 ⇒ (3t + 4)(2t + 1) = 0 ⇒
a
a
a
t 4 1+t 2+t t(1+t)(2+t)
2
2 + 1 + 3 =0 ⇒ 6t +11t+4 =0 ⇒ 6t + 11t +4 = 0 ⇒ (3t + 4)(2t + 1) = 0 ⇒ t = − or t = − . When t = − , then
t = − 4
2
1
4
3
t 1+t 2+t t(1+t)(2+t) 3 2 3
4
t = −
3 4 4 1 1 1
log x = − ⇒ x = a − 3 . When t = − , then log x = − ⇒ x = a − 2 . It is not difficult
a
a
2
3
2
4 1
to verify that x = a − 3 ,x = a − 2 are roots of the original equation.
n
2.94 The coefficients of the last three terms of the expansion of (x lg x + 1) are
2
y
positive integer roots of the equation 3 · 9 −10y · 81 −11 =1. The middle term of the expansion
√

is the root of the equation 3 m =0.1 −2 + 2m , find the value of x .
2

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