Properties of Logarithms

Description/Explanation/Highlights

Video Description

This video explains the properties of logarithms and gives examples of their uses.

Steps and Key Points to Remember

To apply properties of logarithms, follow these steps.

  1. Properties of lgarithms are used to rewrite logarithmic expressions to make solving easier. There are four properties of Logs that will be presented here.
  2. When two numbers are multiplied together inside of a single Log, the factors can be separated into separate Logs that are added. i.e. \(\log_b(x\cdot y)=\log_b x+\log_b y\)
  3. The properties work in both ways. i.e. \(\log(4)+\log(3)=\log(12)\) AND \(\log(12)=\log(4\cdot 3)=\log(4)+\log(3)\) (note: when no base is given the base is assumed to be 10, a common log).
  4. When two numbers are divided inside of a single log, the numbers can be separated into two logs where the log of the denominator is subtracted from the log of the numerator. For example: \(\log_b(x/y)=\log_bx-\log_by\)
  5. As with the previous property, it works in both directions: \(\log(100)-\log(20)=\log(100/20)=\log(5)\)
  6. When the argument of a log is raised to a power, the power can be moved to the front of the log and multiplied. i.e. \(\log_b(x)^y=y\cdot \log_b(x)\) so, \(\log_3(9)^4=4\cdot \log_3(9)=4\cdot 2=8.\)
  7. If the base and the argument are the same, the log is equal to the power. i.e. \(\log_b(b)^x=x\) so, \(\log_3(3)^2=2\)
  8. The \(\log(10)^3=3\) because since there is no base given, it is assumed to be 10, a common log, and therefore the base and the argument are the same.
  9. Knowing the value of a log along with use of these properties can help us solve problems we might otherwise be unable to solve without a calculator. For example if we know that \(\log(2)\) is approximately .3, then we can find the \(\log(4)\) by realizing that the factors of \(4\) are \(2\cdot 2\) which  means the \(\log(2\cdot 2)=\log(2)+\log(2)=.3+.3=.6\).
  10. We can also use the properties to write logs into a single expression such as: \(\log_4(6x+4)-\log_4(2)\) can be rewritten as \(\log_4(\frac{6x+4}{2})=\log_4(3x+2)\)

Here are some key points to keep in mind when applying properties of logarithms.

  • Multiplication inside a single Log becomes addition of two Logs.
  • Division inside a single Log becomes subtraction of two Logs.
  • When a single Log is raised to a power, the power can be moved to the front of the Log and multiplied.
  • If the base and argument of a Log are the same, the Log is equal to the power to which the Log is raised. if there is no power, it is assumed to be 1 and Log = 1.
  • If we know the value of a Log, we can often apply properties to find the value of other Logs.
  • Log properties are also often used to simplify Logs and write multiple Logs into a single Log.

Video Highlights

  • 00:00 Introduction
  • 00:18 \(\log_b(x\cdot y)=\log_b(x)+log_b(y)\) Log of a product property. 
  • 00:45 \(\log(4)+\log(3)=\log(12)\) example of using the Log of a product property
  • 01:30 \(\log_2 (12)=\log_2(3)+\log_2(4)\) example using Log of a product property
  • 02:00 \(\log_b(x/y)=\log_b(x)-\log_b(y)\)
  • 02:47 \(\log(100)-\log(20)=\log(5)\) example of using Log of quotients property.
  • 03:30 \(\log_b(x)^y=y\cdot \log_b(x)\)Log of a power property
  • 03:48 \(\log_3(9)^4=4\cdot \log_3(9)\) example of Log of a power property
  • 04:50 \(\log_b(b)^x=x\) Log property when the base and argument are the same
  • 05:12 \(\log_3(3)^2=2\) example of Log when the base and argument are the same
  • 05:25 \(\log(10)^3=3\) example of common Log when the base and argument are the same
  • 05:49 \(\log_2(2)=1\) example of when the base and argument are the same with a power of 1 implied.
  • 06:10 Using \(\log(2)=.3 \) to find the log(4) example using properties of logs
  • 08:25 Rewrite \(\log_4(6x+4)-\log_4(2)\) as a single expression example of using Log properties to simplify a logarithmic expression

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Related Videos

Rewriting Logs to Exponential Form
Solving Log Equations by Rewriting in Exponential Form
Using Properties of Logs to Solve Log Equations
Solving Logs of All Bases with a Calculator
Solving Log Equations with a Calculator
Using the Inverse to Graph the Log Parent Function
Graphing More Complex Log Functions
Finding Domain and Range of Log Functions

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