Question

Решить уравнение (Log 5 x)^1 + 0,5log5x^2 =6

Answer 1

$log_{5}(x) + log_{5}(x) = 6 \\ log_{5}( {x}^{2} ) = 6 \\2 log_{5}(x) = 6 \\ log_{5}(x) = 3 \\ x = {5}^{3} = 125$

Answer 2

To solve the equation, let's first clarify the expression:

1. (Log 5 x)^1 can be simplified to just Log 5 x since any number raised to the power of 1 is itself.

2. 0.5log5x^2 can be simplified as well. Using the property of logarithms, log_b(x^a) = a*log_b(x), we get: 0.5log5x^2 = log5(x^0.5)

Now, the equation becomes: Log 5 x + log5(x^0.5) = 6

Now, use the logarithmic property log_b(x) + log_b(y) = log_b(xy): Log 5 x + log5(x^0.5) = log5(x * x^0.5)

Next, use another logarithmic property log_b(x^a) = a*log_b(x): log5(x * x^0.5) = log5(x^(1 + 0.5))

Simplify the exponent on the right side: log5(x * x^0.5) = log5(x^1.5)

Now, the equation becomes: log5(x^1.5) = 6

To solve for x, eliminate the logarithm: x^1.5 = 5^6

Now, solve for x by taking both sides to the power of 2/3 (or 1/1.5): x = (5^6)^(2/3) x = 5^(6 * 2/3) x = 5^(4)

Now, calculate the value of x: x = 625

Therefore, the solution to the equation is x = 625.