Question

Log6(x-1)-log6(x+4)=2

Answer 1

Log6(x-1)-log6(x+4)=2
ОДЗ х>1 х> -4

Log6(x-1)/(x+4)=2
(x-1)/(x+4)=6²
 x-1=36*(x+4)
 x-1=36x+144
- 35х= 145
 x= - 145/35 < 0 не подходит под ОДЗ, решения нет

Answer 2

Log₆(x - 1) - log₆(x + 4) = 2
Условие на существование логарифмов:
$\left \{ {{x \ \textgreater \ 1, } \atop {x \ \textgreater \ -4; }} \right.$ ⇒ x > 1. 
По свойству логарифма: logₐb - logₐc = logₐ $\frac{b}{c}$. В нашем случае: 
log₆(x - 1) - log₆(x + 4) = log₆ $\frac{x - 1}{x + 4}$
То есть переходим к такому уравнению при x > 1: 
log₆ $\frac{x - 1}{x + 4}$ = 2
Представим 2 в виде log₆36. 
log₆ $\frac{x - 1}{x + 4}$ = log₆36. 
Т.к. логарифмическая функция принимает каждое свое значение единожды, 
$\frac{x - 1}{x + 4} = 36 \\ x - 1 = 36x + 144 \\ 35x = -145 \\ x = \frac{-29}{7}.$
Вспомним, что x > 1 ⇒ решений нет. 

Ответ: нет решений. 

Answer 3

To solve the equation log6(x-1) - log6(x+4) = 2, we will use logarithmic properties to simplify the equation and find the value of x.

Step 1: Combine the logarithms using the quotient rule of logarithms: log6[(x-1)/(x+4)] = 2

Step 2: Rewrite the equation in exponential form: 6^2 = (x-1)/(x+4)

Step 3: Simplify the equation: 36 = (x-1)/(x+4)

Step 4: Cross-multiply: 36(x + 4) = x - 1

Step 5: Distribute and move all x terms to one side of the equation: 36x + 144 = x - 1

Step 6: Move the constant term to the other side: 36x - x = -1 - 144

Step 7: Combine like terms: 35x = -145

Step 8: Solve for x: x = -145/35

Now, we can simplify the fraction:

x ≈ -4.14 (rounded to two decimal places)

So, the approximate value of x that satisfies the equation is -4.14. Remember that we should always check if the solution is valid by ensuring that the logarithm expressions are positive. Since the argument of the logarithms must be greater than 0, the solution x = -4.14 is valid for this equation.