Solving Exponential Equations

In this section, we will make use of what we have learned about exponential functions to solve equations.

Playlist on Solving Exponential Equations

Make use of the above property if you are able to express both sides of the equation in terms of the same base.

   Step 1: Express both sides in terms of the same base.
   Step 2: Equate the exponents.
   Step 3: Solve the resulting equation.

Solve.

It is not always the case that we will be able to express both sides of an equation in terms of the same base.  For this reason we will make use of the following property.

Make use of the above property if you are unable to express both sides of the equation in terms of the same base.

   Step 1: Isolate the exponential and then apply the logarithm to both sides.
   Step 2: Apply the power rule for logarithms and write the exponent as a factor of the base.
   Step 3: Solve the resulting equation.

Solve.

 

When solving exponential equations and using the above process, the rule of thumb is to choose the common logarithm unless the equation involves the natural base e.   We choose these because there is a button for them on the calculator. However, we could certainly choose any base that we wish; this is the basis for the derivation of the change of base formula.

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Polynomial and Rational Inequalities

Up to this point we have been solving quadratic inequalities.  The technique involving sign charts extends to solving polynomial inequalities of higher degree.

Solving Quadratic, Rational, and Polynomial Inequalities Playlist

Solve.

   Step 1: Determine the critical numbers, which are the roots or zeros in the case of a polynomial inequality.

   Step 2: Create a sign chart.

   Step 3: Use the sign chart to answer the question.

 

 

The last problem shows that not all sign charts will alternate. Do not take any shortcuts and test each interval.

Rational inequalities are solved using the same technique.  The only difference is in the critical numbers.  It turns out that the y-values may change from positive to negative at a restriction. So we will include the zeros of the denominator in our list of critical numbers.

Solve.

 

Tip: Always use open dots for critical numbers that are also zeros of the denominator, or restrictions. This reminds us that they are restrictions and should not be included in the solution set even if the inequality is inclusive.

 

 

 

Use open dots for all of the critical numbers when a strict inequality is involved.

 

 

  

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