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))`.