Question

Log²4(x+5)+3log4(x+5)=2

Answer 1

Log² 4(x+5)+3log4(x+5)=2
Пусть log4(x+5)=t, тогда
t²+3t-2=0
D=9-4*1*(-2)=1
t1= -3+1/2= -1
t2=-3-1/2=-2

Answer 2

To solve the equation $\log_2^4(x+5) + 3\log_4(x+5) = 2$, we can use logarithmic properties to simplify it and solve for $x$:

Let's start by converting all the logarithms to the same base. In this case, let's choose base 2:

$\log_2^4(x+5) + 3\log_4(x+5) = 2$

Now, we can use the change of base formula for logarithms:

$\log_4(x+5) = \frac{\log_2^4(x+5)}{\log_2^4(4)}$

Simplify the denominator:

$\log_4(x+5) = \frac{1}{2} \cdot \log_2(x+5)$

Now, rewrite the equation with the same base:

$\frac{1}{2} \cdot \log_2(x+5) + 3\log_4(x+5) = 2$

Let's substitute $u = \log_2(x+5)$, which simplifies the equation:

$\frac{1}{2}u + 3 \cdot \frac{u}{2} = 2$

$\frac{1}{2}u + \frac{3}{2}u = 2$

Now combine like terms:

$\frac{5}{2}u = 2$

Multiply both sides by 2/5:

$u = \frac{4}{5}$

Substitute back $u = \log_2(x+5)$:

$\log_2(x+5) = \frac{4}{5}$

Now, exponentiate both sides using base 2:

$x+5 = 2^{4/5}$

$x+5 = 2^{0.8}$

$x+5 \approx 1.7411$

Now, solve for $x$:

$x \approx 1.7411 - 5$

$x \approx -3.2589$

So, the approximate solution for the equation is $x \approx -3.2589$.