To provide a solution, first and foremost, we need to use a property of logarithms that says the sum of two logarithms is equal to the logarithm of the multiplication of these two logarithms. Therefore, we can write:
log10(x) + log10(x^5) = 2
as
log10(x*x^5) = 2
Simplified further, we get
log10(x^6) = 2
To get rid of the logarithm, we remove it by making use of the rule a = logb(c) is equivalent to b^a = c, so:
x^6 = 10^2
x^6 = 100
The next step is to take the sixth root of both sides:
x = 100^(1/6)
Solving this, we find the possible solutions for x in the real and complex number space. Therefore, the results are:
x = 10^(1/3) --> This is the real root.
And four complex roots:
x = 10^(1/3)*(-1 - sqrt(3)*i)/2
x = 10^(1/3)*(-1 + sqrt(3)*i)/2
x = 10^(1/3)*(1 - sqrt(3)*i)/2
x = 10^(1/3)*(1 + sqrt(3)*i)/2
Therefore, the equation has one real root and four complex roots.