Showing posts with label logarithm. Show all posts
Showing posts with label logarithm. Show all posts

Monday, November 19, 2012

Solving Exponential Equations


In this section, we will make use of what we have learned about exponential functions to solve 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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Saturday, November 17, 2012

Properties of the Logarithm

The following properties of the logarithm are derived from the rules of exponents.
The properties that follow below are derived from the fact that the logarithm is defined as the inverse of the corresponding exponential.

To familiarize ourselves with the properties we will first expand the following logarithms. (Assume all variables are positive.)


Expand.
Notice that there is no explicit property that allows us to work with nth roots within the argument of the logarithm.  To simplify these, first change the root to a rational exponent then apply the power rule.
When expanding, notice that we must use the same base throughout the expression. For the next set of problems we will first use the properties to expand then substitute in the appropriate values as the last step.
Evaluate
Expanding is useful for learning the rules and properties associated with logarithms but as it turns out, in practice, condensing down to a single logarithm is the skill that we really need to focus on.

Rewrite as a single logarithm (condense).
Tip: When simplifying these down to one logarithm use only one operation at a time and work from left to right. Combining or skipping steps usually leads to mistakes. Do not race, work slowly, and pay attention to notation.
Evaluate (Round to the nearest ten thousandths where appropriate).
Simplify.
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