Note:

**Solve for x in the following equation.**

Example 1:

The above equation is valid only if all the terms in the equation are valid. The term is valid if the term is valid if and the term is valid if The domain is the set of real numbers that are greater than 2, greater than 4, and greater than 1. The domain is therefore the set of real numbers greater than 4.

This is a complex equation because all the bases are different. Let's solve it first by graphing; we do this by changing all the bases to either 10 or e. Why do we change the bases to either 10 or e? Because most calculators have these functions.

Change the original equation into an equation where all the logarithmic
terms have base e.

Rewrite the equation as

Let's call the left side of the equation *f*(*x*) and the right side of the
equation *g*(*x*).

Then
and
Graph *f*(*x*) and *g*(*x*). We are looking for the
point(s) of intersection,
The solution, if any, will
be the value of x in the point(s) of intersection.

The graph of the right side of the equation is the set of points where the
value of y equals zero. This is easy; it is just the x-axis. We then look to
see where the graph of *f*(*x*) crosses the x-axis.

Note that the graph only appears to the right of *x*=4. This is
consistent with our finding that the domain of the original equation is the
set of real numbers greater than 4.

The solution(s) to the original equation is the set of real numbers where *f*(*x*)
crosses the x-axis. (The x-intercepts are the solutions to the problem.)
You will note from the graph that *f*(*x*) crosses the x-axis at about 5.1797. This
means that the equation has one real solution at *x*=5.1797.

The algebraic solution is too difficult for a beginning student. Clearly, graphing is the easiest way to determine the answer.

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