Question

Log3^2(x-3)-log3(x-3)=0

Answer 1

Ответ:

4

Пошаговое объяснение:

Решение на фото

Answer 2

To solve the equation log₃²(x - 3) - log₃(x - 3) = 0, we can use some properties of logarithms. First, we need to recall the following logarithmic rule:

logₐ(b) - logₐ(c) = logₐ(b/c)

Applying this rule to the given equation, we get:

log₃²(x - 3) - log₃(x - 3) = log₃²(x - 3) / log₃(x - 3)

Now, since logₐ(a)² = 2logₐ(a), we can rewrite the equation as:

2log₃(x - 3) - log₃(x - 3) = log₃²(x - 3) / log₃(x - 3)

Next, let's combine the logarithms on the left side:

log₃(x - 3) = log₃²(x - 3) / log₃(x - 3)

Now, we can eliminate the common logarithm on both sides by raising both sides as exponents with base 3:

3^(log₃(x - 3)) = 3^(log₃²(x - 3) / log₃(x - 3))

The bases (3) on both sides are the same, so the exponents must be equal:

log₃(x - 3) = log₃²(x - 3) / log₃(x - 3)

Now, to further simplify, multiply both sides by log₃(x - 3):

log₃(x - 3) * log₃(x - 3) = log₃²(x - 3)

Now, we have a quadratic equation in terms of log₃(x - 3). Let's make a substitution:

Let y = log₃(x - 3)

Then, our equation becomes:

y * y = y²

Solving for y:

y² - y² = 0

0 = 0

Since 0 = 0 is an identity, it means that y can be any real number. Now, let's substitute back y = log₃(x - 3):

log₃(x - 3) = y

Since y can be any real number, log₃(x - 3) can be any real number.

In general terms, the solution to the original equation log₃²(x - 3) - log₃(x - 3) = 0 is:

x - 3 > 0 (because logarithms are only defined for positive values) x - 3 = 3^y, where y is any real number

Therefore, the solution for x is:

x = 3^y + 3, where y is any real number.