## Exponential Equations

Exponential equations contain terms such as 3^{x} or e^{3x} . To solve the equation we find x. There may be no solution for x, one solution or more than one. Often we may substitute for x to simplify, solve the simplified equation then use the substitution to find x. If the equation is simple, we only need to make x the subject or solve by inspection.

Example: Solve

By writingwe haveby identifying powers. It is easily seen now that

Example: Solve

Since the base is the same on both sides (it is equal to 3), we can equate the powers to give

If the bases are not the same but are related, we may be able to make them the same.

Example: Solve

We can use one of the indices laws to writethen the equation becomes

The base is the same on both sides so we can equate the bases, obtainingThis can easily be solved:

If the bases are not related, we can still solve the equation, but we must take logs.

Example: Solve

Taking logs gives

Expand the brackets and collect coefficients of

Now divide bythe coefficient ofto give

This can be evaluated by calculator, but this does not give an exact answer. We can instead make a single log of numerator and denominator, obtaining.

Example: Solve

This has no solutions. We can raise a real number to a negative power, but the result, if the power is real, can never be a negative number.

Example: Solve

Substitutethenand we obtain

We can factorise this to obtainthenor

Use the substitution now to obtainor