Question

Lg^2(100x) - lg^2(10x) + lg^2x = 6

Answer 1

Ответ:

Вроде как х=$\frac{32}{5}$

Answer 2

To solve the given equation:

lg^2(100x) - lg^2(10x) + lg^2x = 6

Let's simplify the equation step by step:

First, recall that log properties state:

1. lg(a * b) = lg(a) + lg(b)
2. lg(a / b) = lg(a) - lg(b)
3. lg(a^b) = b * lg(a)

Using these properties, we can rewrite the equation:

lg^2(100x) - lg^2(10x) + lg^2x = 6

= (lg(100x))^2 - (lg(10x))^2 + (lg(x))^2 = 6

Now, let's simplify the logarithms:

= (2lg(10x))^2 - (lg(10x))^2 + (lg(x))^2 = 6

= 4(lg(10x))^2 - (lg(10x))^2 + (lg(x))^2 = 6

= 3(lg(10x))^2 + (lg(x))^2 = 6

Now, we can substitute the base 10 logarithm with the natural logarithm using the change of base formula:

log_a(b) = log_c(b) / log_c(a)

Let's use the natural logarithm (ln) as the base:

= 3(ln(10x) / ln(10))^2 + (ln(x) / ln(10))^2 = 6

= 3(ln(10x))^2 / (ln(10))^2 + (ln(x))^2 / (ln(10))^2 = 6

= 3(ln(10x))^2 / ln(10)^2 + (ln(x))^2 / ln(10)^2 = 6

Now, we can simplify further by dividing by ln(10)^2:

Let's denote y = ln(10x):

= 3(y^2 / ln(10)^2) + (ln(x))^2 / ln(10)^2 = 6

= 3(y^2 + (ln(x))^2) / ln(10)^2 = 6

Multiplying both sides by ln(10)^2:

= 3(y^2 + (ln(x))^2) = 6 * ln(10)^2

= y^2 + (ln(x))^2 = 2 * ln(10)^2

Now, substitute back y = ln(10x):

= (ln(10x))^2 + (ln(x))^2 = 2 * ln(10)^2

We can't simplify this further without knowing the specific values of x.

If you have specific values for x, you can solve the equation by substituting them in and solving for ln(10x).