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# solve the equation log2 x + logx 2 =2.5

solve equation log2x + logx2=2.5

### 3 Answers by Expert Tutors

Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
5.0 5.0 (3 lesson ratings) (3)
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First change the base on logx2 using the change of base identity:
logx2=log22/log2x=1/log2x
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
Robert J. | Certified High School AP Calculus and Physics TeacherCertified High School AP Calculus and Ph...
4.6 4.6 (13 lesson ratings) (13)
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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

Felice R. | Need Help in Math and Science - Engineer willing to helpNeed Help in Math and Science - Engineer...
4.0 4.0 (1 lesson ratings) (1)
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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