# Exponential Logarithmic Equations

Example. Solve equation x^(1-text(lg)(x))=0.01.

Domain of the equation is x>0.

In this domain both left and right parts take only positive values, therefore we can take logarithm of both sides: text(lg)(x^(1-text(lg)(x)))=text(lg)(0.01) or (1-text(lg)(x))text(lg)(x)=-2.

Now, let y=text(lg)(x) then equation can be rewritten as (1-y)y=-2 or y^2-y-2=0. This equation has two roots: y=-1,y=2.

Therefore, we obtained set of equations: text(lg)(x)=-1,text(lg)(x)=2.

First equation has root x=0.1, second equation has root x=100. All these roots are in domain of the equation and, thus, roots of initial equation.

Here we applied method of taking logartihms of both sides, i.e. we transform equation f(x)=g(x) into equation log_a(f(x))=log_a(g(x)).