## Exponential Equations

Exponential equations contain terms such as 3x or e3x . 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 writing we have by 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 write then the equation becomes The base is the same on both sides so we can equate the bases, obtaining This 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 by the coefficient of to 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 Substitute then and we obtain We can factorise this to obtain then or Use the substitution now to obtain or  