Question

Log3(21-x)=log3(3+x)+1

Answer 1

$log_{3}(21 - x) = log_{3}(3 + x) + log_{3}(3) \\ log_{3}(21 - x) = log_{3}(3 \times (3 + x)) \\ log_{3}(21 - x) = log_{3}(9 + 3x) \\ 21 - x = 9 + 3x \\ 3x + x = 21 - 9 \\ 4x = 12 \\ x = 3$
ОДЗ: см. приложение.

x = 3 -- удовл. ОДЗ => является корнем.

Ответ: 3.

Answer 2

To solve the equation log3(21-x) = log3(3+x) + 1, we can use the properties of logarithms. The key property we'll use is:

log_a(b) = c if and only if a^c = b.

Using this property, we can rewrite the equation as:

3^(log3(21-x)) = 3^(log3(3+x) + 1).

Now, since the bases on both sides are the same (both are 3), we can set the exponents equal to each other:

log3(21-x) = log3(3+x) + 1.

Now, we need to eliminate the logarithms. Recall another property of logarithms:

log_a(b) + log_a(c) = log_a(b * c).

Using this property, we can rewrite the equation again:

log3(21-x) = log3((3+x) * 3).

Now, we know that the logarithm of a number to a certain base is equal to a certain exponent if and only if the number and the base are equal. So, we can set the expressions inside the logarithms equal to each other:

21 - x = (3 + x) * 3.

Now, let's solve for x:

21 - x = 9 + 3x.

To isolate x on one side of the equation, we'll move all the x terms to the left side and the constant terms to the right side:

-x - 3x = 9 - 21.

Combine like terms:

-4x = -12.

Now, divide both sides by -4 to solve for x:

x = -12 / -4.

x = 3.

So, the solution to the equation is x = 3. We should always check the solution to avoid potential extraneous solutions:

log3(21 - 3) = log3(18) = 2, log3(3 + 3) + 1 = log3(6) + 1 = 1 + 1 = 2.

Both sides are equal to 2, so x = 3 is indeed the correct solution.