Question

1+log3(x^4+25)=log√3(√30x^2+12)​

Answer 1

Ответ:

Пошаговое объяснение:

log3 3+log3 (x^4+25)=2log3 (V30x^2+12),   log3 (3x^4+75)=log3 (30x^2+12),

3x^4+75=30x^2+12,  3x^4-30x^2+63=0,  x^4-10x^2+21=0,  x^2=7  или  x^2=3,

x=+-V7  или  x=+-V3,   ОДЗ- х-любое

Answer 2

To solve the equation 1 + log3(x^4 + 25) = log√3(√30x^2 + 12), we'll first simplify the equation by applying the properties of logarithms. Then we'll isolate the variable x and solve for its value.

Let's begin:

1 + log3(x^4 + 25) = log√3(√30x^2 + 12)

Using the change-of-base formula, we can convert the logarithm with base √3 to base 3:

1 + log3(x^4 + 25) = log3(√30x^2 + 12) / log3(√3)

Simplifying the denominator, we have:

1 + log3(x^4 + 25) = log3(√30x^2 + 12) / (1/2)

Applying the property of logarithms that states log_b(x^n) = n * log_b(x), we can simplify further:

1 + log3(x^4 + 25) = 2 * log3(√30x^2 + 12)

Now, let's get rid of the logarithms:

log3(x^4 + 25) = 2 * log3(√30x^2 + 12) - 1

Using another property of logarithms, log_b(x) = y is equivalent to b^y = x:

x^4 + 25 = (√30x^2 + 12)^2

Expanding the square on the right side:

x^4 + 25 = 30x^2 + 2 * √30x^2 * 12 + 12^2

Simplifying:

x^4 + 25 = 30x^2 + 24√30x^2 + 144

Rearranging the equation:

x^4 - 30x^2 - 24√30x^2 + 119 = 0

Now, we can try to factorize the equation, but it seems difficult to factor it directly. In this case, we can use numerical methods or approximation techniques to find an approximate solution.