Question

Log2/3x-4 log3 x+3 = 0

Answer 1

Ответ:

log3 ^2 (x) -4 * log3 (x) + 3 = 0;

Пусть log3 x = a, тогда получим квадратное уравнение:

a^2 - 4 * a + 3 = 0;

Найдем дискриминант квадратного уравнения:

D = b2 - 4 * a * c = (-4)2 - 4 * 1 * 3 = 16 - 12 = 4;

Так как дискриминант больше нуля то, квадратное уравнение имеет два действительных корня:

a1 = (4 - √4)/(2 * 1) = (4 - 2)/2 = 2/2 = 1;

a2 = (4 + √4)/(2 * 1) = (4 + 2)/2 = 6/2 = 3;

1) log3 x = 1;

x = 3^1;

x = 3;

2) log3 x = 3;

x = 3^3;

x = 3 * 3 * 3;

x = 9 * 3;

x = 27;

Ответ: х = 3 и х = 27.

Answer 2

To solve the equation log2/3(x) - 4 * log3(x + 3) = 0, you can use logarithmic properties and algebraic manipulation. Here's how you can solve it step by step:

1. Apply the properties of logarithms to combine the terms on the left side of the equation. Use the fact that log(a) - log(b) = log(a/b) and log(a)^n = n * log(a):

log2/3(x) - log3(x + 3)^4 = 0

2. Use the change of base formula to express both logarithms with a common base, such as base 10:

log(x)/log(2/3) - log((x + 3)^4)/log(3) = 0

3. Now, the equation can be simplified to:

log(x)/log(2/3) - 4 * log(x + 3)/log(3) = 0

4. To further simplify, you can combine the fractions:

[log(x) * log(3) - 4 * log(x + 3) * log(2/3)] / (log(2/3) * log(3)) = 0

5. Set the numerator equal to zero:

log(x) * log(3) - 4 * log(x + 3) * log(2/3) = 0

6. Now, isolate the logarithmic terms:

log(x) * log(3) = 4 * log(x + 3) * log(2/3)

7. Divide both sides by log(3):

log(x) = 4 * (log(x + 3) * log(2/3)) / log(3)

8. To isolate x, you can raise both sides as exponents with base 10 (since we're using common logarithms):

x = 10^[4 * (log(x + 3) * log(2/3)) / log(3)]

This is the solution for the equation. However, it's not possible to simplify it further without knowing the actual values of log(2/3), log(3), and log(2) in the base 10 logarithm. You would need to use a calculator or a computer software to compute the exact value of x based on these logarithmic values.