#1**+1 **

Solve for x:

2 = -(20 log(x^2 + 1))/(log(10))

2 = -(20 log(x^2 + 1))/(log(10)) is equivalent to -(20 log(x^2 + 1))/(log(10)) = 2:

-(20 log(x^2 + 1))/(log(10)) = 2

Divide both sides by -20/(log(10)):

log(x^2 + 1) = -(log(10))/10

-(log(10))/10 = log(1/10^(1/10)):

log(x^2 + 1) = log(1/10^(1/10))

Cancel logarithms by taking exp of both sides:

x^2 + 1 = 1/10^(1/10)

Subtract 1 from both sides:

x^2 = 1/10^(1/10) - 1

Take the square root of both sides:

**Answer: | x = sqrt(1/10^(1/10) - 1) or x = -sqrt(1/10^(1/10) - 1)**

Guest Apr 5, 2017

#2**+1 **

2 = -20 log_{10}(1/1+x^2) divide both sides by -20

-1/10 = log_{10}(1/1+x^2)

This says that

10^(-1/10) = 1 / [ 1 + x^2 ] exponentiate both sides to -10

10 = [ 1 / ( 1 + x^2)]^(-10) and we can write

10 = [ (1 + x^2) ] ^(10) take each side to the 1/10 power

10^(1/10) = 1 + x^2 subtract 1 from both sides

10^(1/10) - 1 = x^2 take both roots

x = ± sqrt [10^(1/10) - 1] ≈ ± 0.50885

CPhill
Apr 5, 2017