Question

Log4(x-1)=log 4(x²-x-16)

Answer 1

Ответ:

x-1 = x²-x-16

x²-2x-15=0

x1=5 или x2=-3( не является решением)

Ответ: 5

Пошаговое объяснение:

Answer 2

To solve the equation log₄(x - 1) = log₄(x² - x - 16), we can use the property of logarithms that states:

logₐ(b) = logₐ(c) if and only if b = c.

Using this property, we can set the expressions inside the logarithms equal to each other:

x - 1 = x² - x - 16

Now, let's solve for x:

Step 1: Move all terms to one side of the equation to get it into quadratic form: x² - x - 16 - (x - 1) = 0

Step 2: Combine like terms: x² - x - x + 16 - 1 = 0

Step 3: Simplify the equation: x² - 2x + 15 = 0

Step 4: Factor the quadratic equation: (x - 5)(x - 3) = 0

Step 5: Use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero: x - 5 = 0 or x - 3 = 0

Step 6: Solve for x: x = 5 or x = 3

However, we need to check if these solutions satisfy the original equation. Let's verify them one by one:

For x = 5: log₄(5 - 1) = log₄(5² - 5 - 16) log₄(4) = log₄(25 - 5 - 16) log₄(4) = log₄(4) Both sides are equal, so x = 5 is a valid solution.

For x = 3: log₄(3 - 1) = log₄(3² - 3 - 16) log₄(2) = log₄(9 - 3 - 16) log₄(2) = log₄(-10)

We encounter a problem here since the logarithm of a negative number is undefined in the real number system. Therefore, x = 3 is not a valid solution.

The only valid solution for the equation log₄(x - 1) = log₄(x² - x - 16) is x = 5.