Rewriting Logarithms to Exponential Form

Description/Explanation/Highlights

Video Description

This video explains what a logarithm is and how to rewrite it to exponential form.

Steps and Key Points to Remember

To change from exponential to logarithmic form and from logarithmic to exponential form, follow these steps:

  1. Logarithms (LOGS) are used to find missing values that occur in the exponent of an equation. Because of this, it is very important to understand the relationship between exponential and logarithmic forms of an equation.
  2. To write the example \(3^x=27\) in logarithmic form, we write 3 as the base of the Log, 27 as the argument and x as the missing exponent on the other side of the equal sign. So in log form, it will look like this :  \(\log_3 27=x\)
  3. We know that 3 to the third power is 27, so \(\log_3 27=3\). The missing exponent is 3. Remember, logs are equal to the missing exponent.
  4. To find the \(\log_2 16\), we would re-write it in exponent form to see if we can determine the missing exponent. In this case it is \(2^x=16\) (x is the missing exponent). Therefore, since we know that \(2^4=16\), then \(\log_2 16=4\).
  5. Sometimes re-writing won’t give us an exact answer, but we could still use it to estimate. For example to solve \(\log_5 17=x\) for x, we would re-write to \(5^x=17\). We can’t find the exact answer without a table or calculator, but we know it’s somewhere between 1 and 2.
  6. Sometimes the missing variable is somewhere besides the exponent but we need to re-write as a log. Use the same rules. For example: \(6^3=x\) rewrites to \(\log_6 x=3\)
  7. You will sometimes see a log written without an apparent base. When this occurs, it is called a common log and has an assumed base of 10. For example, \(\log (3)=x\) is rewritten as \(10^x=3\) and is the same thing as writing \(\log_{10} (3)=x\).

Here are some key points to keep in mind when working with logarithmic and exponential forms.

  • In the equation,  \(\log_3 81=4\), 3 is the base, 81 is the argument, and 4 is the exponent.
  • In the equation, \(3^4=81\), 3 is the base, 4 is the exponent, and 81 is the argument.
  • Common logs are base 10 logs and are often written without a base.

Video Highlights

  • 00:00 Introduction
  • 00:15 \(3^x=27\) rewritten to a log
  • 01:12 \(\log_2 16=x\) example rewritten in exponential form
  • 02:18 \(\log_5 17=x\) example rewritten in exponential form
  • 03:00 \(6^3=x\) example rewritten in logarithmic form
  • 03:40 \(\log (3)=x\) example of rewriting a common logarithm
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Graphing More Complex Log Functions
Finding Domain and Range of Log Functions
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