Solve the logarithmic equation.
Log x +log2x = 4
Wouldn't this be no solution because log x is undefined?
logx + log 2x = 4
log (x(2x)) = 4 , using the rule log AB = logA + logB
log (2x^2) = 4
so 2x^2 = 10^4
2x^2 = 10000
x^2= 5000
x = ± √ 5000
but x > 0 for log x to be defined
so x = √5000
Actually, the natural logarithm function (log) is defined for positive values of x. So, in this case, we have the logarithmic equation:
log(x) + log(2x) = 4
To solve this equation, we can use the properties of logarithms. Specifically, we can use the logarithm property that states:
log(a) + log(b) = log(ab)
Applying this property to the given equation, we can rewrite it as:
log(2x^2) = 4
Now, we can exponentiate both sides of the equation with base 10 (or any other positive base) to eliminate the logarithm. Exponentiating both sides of the equation gives us:
10^(log(2x^2)) = 10^4
Simplifying further, we have:
2x^2 = 10^4
Solving for x, we can divide both sides by 2:
x^2 = 10^4 / 2
x^2 = 5000
Taking the square root of both sides, we get:
x = ±√5000
Therefore, the solutions to the equation are x = √5000 and x = -√5000.