Question

Log3(x+3)(x+5)+log3 x+3/x+5=4​

Answer 1

Ответ:

$log_{3}(x + 3)(x + 5) + log_{3}( \frac{x + 3}{x + 5} ) = 4 \\ log_{3}(x + 3) + log_{3}(x + 5) + log_{ 3}(x + 3) - log_{3}(x + 5) = 4 \\ log_{3}(x + 3)(x + 3) = 4 \\ {(x + 3)}^{2} = {3}^{4} \\ x + 3 = {3}^{2} \\ x + 3 = 9 \\ x = 9 - 3 \\ x = 6$

Answer 2

To solve the equation log3(x+3)(x+5) + log3(x+3/x+5) = 4, we can use logarithmic properties to simplify it.

1. First, combine the two logarithms on the left side using the product rule of logarithms:

log3[(x+3)(x+5) * (x+3/x+5)] = 4

2. Now, we can eliminate the logarithm by converting the equation to exponential form:

3^(log3[(x+3)(x+5) * (x+3/x+5)]) = 3^4

3. This simplifies to:

(x+3)(x+5) * (x+3/x+5) = 81

4. Next, let's deal with the fraction by multiplying both sides by (x+5):

(x+3)(x+5) * (x+3/x+5) * (x+5) = 81 * (x+5)

5. Simplify:

(x+3)(x+5) * (x+3) = 81 * (x+5)

6. Expand the expressions:

(x^2 + 8x + 15)(x+3) = 81x + 405

7. Distribute and simplify further:

x^3 + 11x^2 + 39x + 45 = 81x + 405

8. Move all terms to one side to set the equation equal to zero:

x^3 + 11x^2 + 39x + 45 - 81x - 405 = 0

9. Combine like terms:

x^3 + 11x^2 - 42x - 360 = 0

10. Now, we need to solve this cubic equation. Unfortunately, solving cubic equations directly can be quite involved. You can try factoring, synthetic division, or numerical methods like the Newton-Raphson method to approximate solutions. In some cases, rational or integer roots may be easier to find, which can help simplify the equation.