Question

Помогите 2log5(x-1)=log5(7-x) нужно подробно расписать решение с объяснением...

Answer 1

Сначала напиши ОДЗ, а потом уже то, что обведено (рано написала XD)

Answer 2

Solving the Equation 2log5(x-1)=log5(7-x)

To solve the equation 2log5(x-1)=log5(7-x), we can use properties of logarithms to simplify and solve for the variable x.

Step 1: Apply the Power Rule of Logarithms

The power rule of logarithms states that log_b(x^n) = n*log_b(x). We can apply this rule to the left side of the equation to simplify it.

2log5(x-1) = log5((x-1)^2)

Step 2: Use the Product Rule of Logarithms

The product rule of logarithms states that log_b(xy) = log_b(x) + log_b(y). We can apply this rule to the right side of the equation to simplify it.

log5((x-1)^2) = log5(7-x)

Step 3: Set the Expressions Equal to Each Other

Now that both sides of the equation are in the form log_b(y), we can set the expressions inside the logarithms equal to each other.

(x-1)^2 = 7-x

Step 4: Solve for x

We can now solve the quadratic equation (x-1)^2 = 7-x to find the values of x that satisfy the original equation.

Expanding and rearranging the equation: x^2 - 2x + 1 = 7 - x x^2 + x - 6 = 0

Now, we can factorize the quadratic equation: (x+3)(x-2) = 0

So, the solutions for x are: x = -3 and x = 2

Conclusion

The solutions to the equation 2log5(x-1)=log5(7-x) are x = -3 and x = 2.

This completes the detailed solution with explanations for the given equation. If you have any further questions or need additional assistance, feel free to ask!