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Solving Exponential and Logarithmic Equations

Here we will make use of what we have learned about exponentials and logarithms to solve equations.

One-to-One Property of Exponential Functions –
If b^n = b^m then n=m

We may make use of the above property if we are able to express both sides of the equation in terms of the same base as shown here.

  Solve Recall the rule of exponent
  Step 1: Express both sides in terms of the same base.
Step 2: Equate the exponents. Step 3: Solve for the variable.

A. 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.

One-to-One Property of Logarithmic Functions – For all real b, b > 0 and b ≠ 1
  if and only if x = y

 

  Solve  
Step 1: Take the common log of both sides. Step 2: Apply the power rule for logarithms
Step 3: Solve for the variable.  
Exact Answer Approximate answer rounded off to the nearest hundredth.

B. Solve

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

  Prove  
Step 1: Apply the definition of the logarithm. Step 2: Take the base a log of both sides.
Step 3: Apply the power rule for logarithms.  
  Step 4: Solve for y.
  For the last step simply equate the equivalent forms of y above.

We can also use the one-to-one property for logarithms to solve logarithmic equations. If we are given an equation with a logarithm of the same base on both sides we may simply equate the arguments.

   
Step 1: Use the rules for logarithms to isolate a log on both sides of the equation.
 
  Step 2: Equate the arguments.
   
Step 3: Solve for the variable.
 
    Step 4: We must check to see if they work.

Be sure to check to see if the solutions that we obtain
solve the original logarithmic equation. In this manual we will put a check mark next to the solution after we determine that it really does solve the equation. This process sometimes results in extraneous solutions so we must check our answers.

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