Question

Lg^2 x + 2lg 100x -7 =0

Answer 1

Lg²  x + 2lg 100x -7 =0

Lg² x + 2(lg 100+lg x) -7 =0

Lg² x + 2( 2+lg x) -7 =0

Lg² x + 4+2lg x -7 =0

Lg² x + 2lg x -3 =0        lg x=a

a²+2a-3=0

D=4+12=16

a=(-2+4)/2=1       lg x=1   x=10

a=(-2-4)/2=-3      lg x= -3  x=1/1000 =0,001

Answer 2

To solve the equation lg^2(x) + 2lg(100x) - 7 = 0, let's break it down step by step.

First, let's simplify the equation by using a logarithmic property: lg(a^b) = b * lg(a). This property states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Using this property, we can rewrite the equation as follows:

2lg(x) + 2lg(10) + 2lg(x) - 7 = 0

Simplifying further:

4lg(x) + 2lg(10) - 7 = 0

Next, let's simplify the expression 2lg(10) using a logarithmic property: lg(10) = 1. This property states that the logarithm of 10 to the base 10 is equal to 1.

Substituting lg(10) = 1 in the equation:

4lg(x) + 2(1) - 7 = 0

4lg(x) + 2 - 7 = 0

4lg(x) - 5 = 0

Now, we can isolate the logarithmic term:

4lg(x) = 5

Dividing both sides of the equation by 4:

lg(x) = 5/4

To eliminate the logarithm, we can rewrite the equation in exponential form. In exponential form, if y = log(base a)(x), then x = a^y.

Using this form, we have:

x = 10^(5/4)

Therefore, x = 10^(1.25) ≈ 17.7828.

So the solution to the equation lg^2(x) + 2lg(100x) - 7 = 0 is x ≈ 17.7828.