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solve the equation log2 x + logx 2 =2.5

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3 Answers

First change the base on logx2 using the change of base identity:
logx2=log22/log2x=1/log2x
Then your equation becomes
log2x + 1/log2x =2.5
Let u=log2x and simplify:
u+1/u=2.5
u²-2.5u+1=0
 
Use the quadratic formula to solve this quadratic equation, get u = 2 or 1/2.
 
For u=2=log2x,  x=4.
For u=1/2=log2x, x=√2
Therefore, the two solutions are x=4 and x=√2
 
Check: log24 + log42 = 2+1/2=2.5
log2√2 + log √22 = 1/2 + 2 =2.5
Use the change of base formula,
log2 x + 1/log2 x = 2.5
Let u = log2 x,
u + 1/u = 2.5
Multiply both sides by 2u, and collect all terms in one side,
2u^2 - 5u + 2 = (2u-1)(u-2) = 0
u = 1/2, x = sqrt(2)
or
u = 2, x = 2^2 = 4
Answer: x = sqrt(2), and x = 4
 
if you remember that if you were going to change base of a log
 
logb(x) = logd(x) / logd(b)
 
if we used this principle with your eqn
 
log2x + logx2 = 2.5
 
we could get the eqn into on base
 
ln(x)/ln(2) + ln(2)/ln(x) = 2.5
 
multiply by ln(2)ln(x) to remove denominator
 
ln2(x) + ln2(2) = 2.5 ln(x)ln(2)
 
ln(2) = 0.693
2.5ln(2) = 1.733
ln2(2) = 0.480
 
 
let y = ln(x) your eqn will become
 
y2 -1.733y + 0.480 = 0
 
Solving for y using quadratic eqn
 
you get
 
y = 1.3869 and y = 0.3461
 
but we need to solve for x
 
x = ey
 
x = 4 and x = 1.4135 = sqrt(2)
 
to verify if you remember
 
logb(mn) = n · logb(m)
 
in our case
 
verify x =4
 
ln(4)/ln(2) + ln(2)/ln(4) = 2.5
 
ln(22)/ln(2) + ln(2)/ln(22) = 2.5
 
2ln(2)/ln(2) +ln(2)/(2ln(2)) = 2 + 1/2 = 2.5
 
verify x = sqrt(2) = 21/2
 
ln(2)/(ln(sqrt(2))) + ln(sqrt(2))/ln(2) =
 
ln(2)/(1/2ln(2) + 1/2ln(2)/ln(2)
 
1/(1/2) + 1/2 = 2 + 1/2 = 2.5