Question

Log4^2 x - log4 x^5 + 4 =0

Answer 1

$log_{4}^2 x - log_{4} x^5 + 4=0 \\ log_{4}^2 x - 5*log_{4} x + 4=0$
ОДЗ: x > 0
пусть  $log_{4}x=a$, тогда
$a^2-5a+4=0 \\ D=25-4*4=9=3^2$ 
$a_1= \frac{5-3}{2}= 1$   или    $a_2= \frac{5+3}{2}= 4$
отсюда:
$log_{4}x_1=1$     или     $log_{4}x_2=4$
$x_1=4$                      $x_2=256$

Answer 2

To solve the equation log₄(x^2) - log₄(x^5) + 4 = 0, you can use the properties of logarithms.

First, let's apply the properties of logarithms to simplify the equation:

log₄(x^2) - log₄(x^5) + 4 = 0

Now, we can use the properties of logarithms to simplify further:

log₄(x^2) - log₄(x^5) + log₄(4) = 0

Now, we can combine the logarithms using the properties of logarithms:

log₄((x^2) / (x^5)) + log₄(4) = 0

Next, use the properties of logarithms to combine the terms inside the logarithm:

log₄((1 / x^3)) + log₄(4) = 0

Now, we have a sum of logarithms, so we can use the properties of logarithms to combine them:

log₄((1 / x^3 * 4)) = 0

Now, we have a single logarithm, and we want to find the value of x. To do this, we can use the property of logarithms that says if logₐ(b) = c, then a^c = b. In this case, a = 4, c = 0, and b = 1 / (x^3 * 4). So:

4^0 = 1 / (x^3 * 4)

Since 4^0 is equal to 1, we have:

1 = 1 / (x^3 * 4)

Now, multiply both sides of the equation by x^3 * 4 to isolate x:

x^3 * 4 = 1

Now, divide both sides by 4 to solve for x^3:

x^3 = 1/4

To find x, take the cube root of both sides:

x = ∛(1/4)

x = 1/∛4

So, the solution to the equation log₄(x^2) - log₄(x^5) + 4 = 0 is:

x = 1/∛4