Question

Log 3x = log 3 (6 - x в квадрате)

Answer 1

log 3x = log 3 (6 - x в квадрате),х∈(0,√6)

х=6-х2

х-6+х2=0

х2+х-6=0

х=-1+-√1+24/2

х=-1+-5/2

х=-1-5/2

х=2

х=-3

х∈(0,√6)

х=2

Answer 2

To solve the equation log(3x) = log3(6 - x^2), we will use the properties of logarithms. The equation states that the logarithm of 3x to the base 10 is equal to the logarithm of (6 - x^2) to the base 3.

First, let's write the equation using logarithm properties:

log₃(3x) = log₃(6 - x²)

Since both sides have the same base (logarithm base 3), we can eliminate the logarithm and equate the expressions inside:

3x = 6 - x²

Now, we have a simple quadratic equation to solve. We can rearrange the equation to get it in the standard form:

x² + 3x - 6 = 0

Now, we can either factor this quadratic equation or use the quadratic formula to find the solutions for x. Let's use the quadratic formula:

For a quadratic equation of the form ax² + bx + c = 0, the solutions are given by:

x = (-b ± √(b² - 4ac)) / 2a

In our case, a = 1, b = 3, and c = -6:

x = (-(3) ± √(3² - 4(1)(-6))) / 2(1)

x = (-3 ± √(9 + 24)) / 2

x = (-3 ± √33) / 2

So, the two solutions for x are:

1. x = (-3 + √33) / 2
2. x = (-3 - √33) / 2

These are the exact solutions for the equation log(3x) = log3(6 - x^2). If you need the approximate decimal values, you can calculate them using a calculator.