Question

2log5(2x)+log5(x-1/x)=log5(8x^2+1/x-3)

Answer 1

ОДЗ
x>0
(1-x)/x>0
8x²+1/x-3>0

(2x)²(1-x)/x=8x²+1/x-3
4x-4x²=8x²+1/x-3
12x²-4x-3+1/x=0 умножим все на х
12х³-4х²-3х+1=0
4х²(3х-1)-(3х-1)=0
(3х-1)(4х²-1)=0
1) 3х-1=0
    x1=1/3
 2) 4x²=1
      x2=1/2
       x3=-1/2 этот корень не подходит к ОДЗ

Answer 2

To solve the equation 2log5(2x) + log5(x-1/x) = log5(8x^2+1/x-3), we can use logarithmic properties to simplify and solve for x. Here's the step-by-step solution:

Step 1: Use logarithmic properties to simplify the equation. First, we'll use the property that log(a) + log(b) = log(a * b) and log(a) - log(b) = log(a / b):

2log5(2x) + log5(x-1/x) = log5(8x^2+1/x-3)

Using the property mentioned above, we can rewrite the equation as:

log5((2x)^2) + log5(x-1/x) = log5(8x^2+1/x-3)

Step 2: Combine the logarithms on the left side of the equation using another logarithmic property: log(a) + log(b) = log(a * b)

log5((2x)^2 * (x-1/x)) = log5(8x^2+1/x-3)

Step 3: Simplify the expression inside the logarithm on the left side of the equation: (2x)^2 * (x-1/x) = (4x^2) * (x-1/x) = 4x^3 - 4

So, the equation becomes:

log5(4x^3 - 4) = log5(8x^2 + 1/x - 3)

Step 4: Remove the logarithms by setting the expressions inside them equal to each other:

4x^3 - 4 = 8x^2 + 1/x - 3

Step 5: Move all the terms to one side to set the equation to zero:

4x^3 - 8x^2 + 1/x - 1 = 0

Step 6: To solve this cubic equation, it might not be easy to find an exact solution. In some cases, numerical methods or graphing calculators might be used to find approximate solutions. In case you're required to approximate the solutions, you can use numerical methods like Newton-Raphson or the bisection method.

However, as an AI language model, I'm unable to perform numerical calculations. If you need an approximate solution to the equation, I recommend using a graphing calculator or numerical solver. Alternatively, if you are working on a specific problem, there might be a simplification or specific range of solutions required that could help narrow down the solutions.