Question

(0,25)^log0,25(2x-1)=x+5​

Answer 1

Ответ:

$x=6$

Объяснение:

$(0,25)^{log_{0,25}(2x-1)}=x+5\\\\2x-1>0\\2x>1\\x>0,5\\\\(0,25)^{log_{0,25}(2x-1)}=x+5\\2x-1=x+5\\2x-x=5+1\\x=6\\6>0,5\\\\x=6$

*** Использована формула: $a^{log_ab}=b$

Answer 2

To solve the equation, let's break it down step by step.

The given equation is:

(0.25)^(log0.25(2x-1)) = x + 5

We'll start by simplifying the left side of the equation using the logarithmic property:

log_a(a^b) = b

So, we have:

(0.25)^[(log(2x-1))/(log0.25)] = x + 5

Now, let's simplify further. The logarithm base of 0.25 can be written as -2 (since 0.25 = 1/4 and log_base_4(1/4) = -2).

(0.25)^[log(2x-1)/log(0.25)] = x + 5

Now, rewrite the left side using exponential notation:

4^(-2 * log(2x-1)) = x + 5

Apply the power rule:

1 / (4^(2 * log(2x-1))) = x + 5

To proceed, let's simplify further by using the property:

a^(b * c) = (a^b)^c

[(4^2)^(log(2x-1))]^(-1) = x + 5

[16^(log(2x-1))]^(-1) = x + 5

Now, rewrite the equation in exponential notation:

[2^(4)]^log(2x-1))^(-1) = x + 5

2^([4 * log(2x-1)] * (-1)) = x + 5

Apply the power rule again:

2^(-4 * log(2x-1)) = x + 5

Now, let's simplify further using the logarithmic property:

log_a(a^b) = b

-4 * log(2x-1) = log(2x-1) * (x + 5)

At this point, we have a logarithmic equation. To solve it, we'll divide both sides by log(2x-1):

-4 = x + 5

Now, isolate the variable x:

x = -4 - 5

x = -9

Therefore, the solution to the equation is x = -9.