Solving Logarithmic Equations by Rewriting in Exponential Form
Description/Explanation/Highlights
Video Description
This video explains how to solve logarithmic equations by rewriting them into exponential form.
Steps and Key Points to Remember
To solve for a missing x in a logarithmic equation, follow these steps.
- To solve for a missing x in the base, argument or exponent of a logarithmic equation, begin by rewriting the equation in exponential form and then solve the exponential equation.
- To solve when the missing x is the exponent, rewrite and find an exponent that will solve the equation. i.e. \(\log_5 (125)=x\) rewrites to \(5^x=125\) (see separate lesson to learn to rewrite). In this case x = 3 will solve the exponential equation.
- If the answer is not obvious after rewriting when the exponent is the missing variable, a table or calculator may be needed to find the log (covered in a separate lesson).
- To solve when the missing x is in the base, rewrite and raise both sides to a reciprocal power to solve for x. For example: \(\log_x 81=4\) rewrites to \(x^4=81\) and raising both sides to the reciprocal power of 4 gives us \((x^4)^{\frac{1}{4}}=81^{\frac{1}{4}}\)
- Remember, \(81^{\frac{1}{4}}\) means \(\sqrt[4]{81}\). Therefore, x = 3.
- Solving for x in the argument is generally the easiest one. For example, rewrite \(\log_2 x=5\) as \(2^5=x\). Solve \(2^5\) and \(x=32\).
- For really big numbers you may need a calculator with an exponent key.
Here are some key points to keep in mind when working with logarithmic and exponential forms.
- Rewrite all LOGs with a missing x into exponential form to solve.
- When the missing x is in the exponent, additional math may be involved if the answer is not obvious. Most of those will require a table or calculator and are covered in other lessons.
- Solving for x in the base requires use of a reciprocal power and using roots and powers to solve. Again, see the separate lesson in the exponents section.
- Solving for x in the argument is the easiest one but may require a calculator for large numbers. Use the ^ on calculators that are equipped. For example, 2 raised to the 5th power is enterd as 2^5.
Video Highlights
- 00:00 Introduction
- 00:14 \(\log_5 (125)=x\) example of solving for a missing exponent
- 01:10 \(\log_x 81=4\) example of solving for a missing base
- 02:28 \(\log_2 x=5\) example of solving for a missing argument
To watch this video on YouTube in a new window with clickable highlights, click here
Related Videos
Rewriting Logs to Exponential Form
Four Properties of Logs
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