Question

Решите уравнение log22 x – 5log2 x + 6=0. Ответ будет ровняться?

Answer 1

Ответ: x₁=4,   x₂=8.

Пошаговое объяснение:

log₂²x-5*log₂x+6=0     ОДЗ: x>0

Пусть log₂x=t   ⇒

t²-5t+6=0    D=1

t₁=2    ⇒    log₂x=2      x₁=2²=4

t₂=3    ⇒   log₂x=3       x₂=2³=8.

Answer 2

To solve the equation log₂(2x) - 5log₂(x) + 6 = 0, we can use logarithm properties to simplify it.

Recall the following logarithm properties:

1. logₐ(a) = 1
2. logₐ(a^b) = b * logₐ(a)
3. logₐ(xy) = logₐ(x) + logₐ(y)

Now, let's use these properties to simplify the equation:

log₂(2x) - 5log₂(x) + 6 = 0

Using property 2: log₂(2x) = log₂(2) + log₂(x) = 1 + log₂(x)

Substitute this into the equation:

1 + log₂(x) - 5log₂(x) + 6 = 0

Now, combine the logarithmic terms:

log₂(x) - 5log₂(x) = -7

Using property 3: log₂(x) - 5log₂(x) = log₂(x) - log₂(x^5) = log₂(x/x^5) = log₂(x^(1-5)) = log₂(x^(-4))

So, we have:

log₂(x^(-4)) = -7

Now, we can rewrite this equation in exponential form:

x^(-4) = 2^(-7)

Now, we can solve for x:

x^(-4) = 1/128

To eliminate the negative exponent, take the reciprocal of both sides:

x^4 = 128

Finally, find the fourth root of both sides to get the value of x:

x = ±√(128) ≈ ±11.3137

Thus, the two solutions for x are approximately x = 11.3137 and x = -11.3137.