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Solving Exponential and Logarithmic Functions




 
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More Lessons for Algebra II

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A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

  • common and natural logarithms
  • change of base formula
  • how to solve exponential equations with different bases
  • how to solve logarithmic equations
  • how to solve logarithmic equations with multiple logs


Common and Natural Logarithms

We can use many bases for a logarithm, but the bases most typically used are the bases of the common logarithm and the natural logarithm. The common logarithm has base 10, and is represented on the calculator as log(x). The natural logarithm has base e, a famous irrational number, and is represented on the calculator by ln(x). The natural and common logarithm can be found throughout Algebra and Calculus.

How to define natural and common logs.

Defines common log, log x, and natural log, ln x, and works through examples and problems using a calculator



Change of Base Formula

When we encounter logarithms with bases not of the common or natural logarithm, we often need the change of base formula. The change of base formula allows us to convert a logarithm from one base to another. By using the change of base formula, we can change a logarithmic term to allow us to input it into a calculator. Most calculators only accept logarithms of base 10 or base e.

This video shows how to determine the value of any logarithm using the change of base formula.

This video shows the change of base formula for logarithms, and a few examples of evaluating logarithms using the formula and a calculator.


 
This video explains how to use the change of base formula for logarithms to evaluate logarithmic expressions. The values are verified by writing an exponential equation using the log values.

Solving Exponential Equations with the Different Bases

Sometimes we are given exponential equations with different bases on the terms. In order to solve these equations we must know logarithms and how to use them with exponentiation. We can access variables within an exponent in exponential equations with different bases by using logarithms and the power rule of logarithms to get rid of the base and have just the exponent.

Solving a Exponential Equation with Different Bases


Using logarithms to solve equations with different bases

Solving Logarithmic Equations

Just as we can use logarithms to access exponents in exponential equations, we can use exponentiation to access the insides of a logarithm. Solving logarithmic equations often involves exponentiating logarithms in order to get rid of the log and access its insides. Sometimes we can use the product rule, the quotient rule, or the power rule of logarithms to help us with solving logarithmic equations.
This video explains how to solve logarithmic equations. 


 
In this video we solve equations where the variable is inside a logarithm.

Solving a Logarithmic Equation with Multiple Logs

When given a problem on solving a logarithmic equation with multiple logs, students should understand how to condense logarithms. By condensing the logarithms, we can create an equation with only one log, and can use methods of exponentiation for solving a logarithmic equation with multiple logs. This requires knowledge of the product, quotient and power rules of logarithms. This video shows how to solve a logarithmic equation using properties of logarithms and some other algebra techniques.


This video shows how to solve a logarithmic equation using properties of logarithms and some other algebra techniques.
Solving Logarithmic Equations


 

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You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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