Solving Quadratic Exponential Equations by Substitution

Some exponential equations can be factorised in linear factors. The simplest can be factorised into quadratic equations. We then put each factor equal to zero and solve it.

Example: Solve(1)

Factorise to get

or

The above equation has two solutions. In general, as for quadratic equations, an exponential which can be expressed as two factors can have one, two or no solutions. It is convenient to make clear the connection by expressing the original equation as a quadratic using the substitutionThenand equation (1) above becomesThis equation factorises to givesoSince the original equation was expressed in terms ofwe still have to findbut we can use the substitutionwith the values ofthat we have found, to find

or

Example: Solve

Substituteto getand factorise this expression to give

This has no solution since there is no real log of a negative number.

Example: Solve

Substituteto getand factorise this expression to give

This has no solution since there is no real log of a negative number.

This has no solution since there is no real log of a negative number hence the equation has no solutions.