Question

Log x-1 (x^2+8x+51)=2

Answer 1

Log (x-1) (x^2+8x+51)=2
ОДЗ по определению логарифма
x-1>0
x-1≠1
x^2+8x+51>0
x>1
x≠2
D=64-4*51<0 всегда выполняется
x∈(1 2) U (2 +∞)
(x-1)² = x²+8x+51
x²-2x+1=x²+8x+51
10x=-50
x=-5 не проходит по ОДЗ
решений нет 

Answer 2

To solve the equation log_x-1 (x^2 + 8x + 51) = 2, we need to understand that the notation "log_x-1" refers to the logarithm with base (x-1). In other words, the equation can be rewritten as:

log(x^2 + 8x + 51) / log(x - 1) = 2

Now, we can solve the equation step by step:

Step 1: Simplify the logarithm on the left-hand side.

log(x^2 + 8x + 51) = 2 * log(x - 1)

Step 2: Use the property of logarithms that states log(a^b) = b * log(a).

log(x^2 + 8x + 51) = log((x - 1)^2)

Step 3: Since the logarithms are equal, the expressions inside them must be equal.

x^2 + 8x + 51 = (x - 1)^2

Step 4: Expand the right-hand side.

x^2 + 8x + 51 = x^2 - 2x + 1

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

x^2 + 8x + 51 - x^2 + 2x - 1 = 0

Step 6: Simplify the equation.

10x + 50 = 0

Step 7: Solve for x.

10x = -50

x = -50 / 10

x = -5

Step 8: Check for validity.

Since the base of the logarithm (x - 1) cannot be negative or zero, x = -5 is not a valid solution. Therefore, there are no real solutions to the equation log_x-1 (x^2 + 8x + 51) = 2.