Solving Exponential Equations with X in the Exponent Using Logarithms
Description/Explanation/Highlights
Video Description
This video explains how to solve for missing variables in the exponent using Logarithms and the change of base property using a calculator.
Steps and Key Points to Remember
To solve an equations with a missing variable in the exponent using logarithms, follow these steps:
- To get a final answer, you will need a calculator with a LOG key.
- You recognize equations that can be solved with logarithms by missing variables in the exponent i.e. \(3(2.4)^x=30\)
- To solve an equation like this, begin by isolating the base and the power with the missing variable. In the example in #2 above, divide both sides of the equation by 3 to get rid of the 3 in front leaving: \((2.4)^x=10\).
- Rewrite the exponential equation into logarithmic form (see separate lesson). In this example the result will be \(\log_{2.4}(10)=x\).
- Use the change of base property to rewrite using common logs. i.e. \(\log(10)/\log(2.4)\)
- Using the calculators’s LOG key, put this in the calculator and divide. The result is x = 2.6301 (rounded to four decimal places).
- Sometimes the process of isolating is more complex and some final Algebra may be required after finding the log as in this example: \(40+30(2.1)^{x-1}=100\).
- To isolate in this example, you must subtract 40 from both sides AND divide by 30 leaving \((2.1)^{x-1}=2\). Note that in this example after isolating, there is still a -1 in the exponent with x. We will take care of this after applying the change of base property to solve.
- Rewriting in logarithmic form gives us: \(x-1=\log_{2.1}(2)\).
- Use the change of base property to rewrite using common logs. i.e. \(x-1=\log(2)/\log(2.1)\)
- Use the calculator LOG key to put this in the calculator and solve. Note that the answer you get will be what \(x-1\) equals. i.e. \(x-1=.93424\)
- To get the final value for x, simply add one to both sides and x = 1.93424.
Here are some key points to keep in mind when solving equations using logarithms.
- To get a final answer, you will need a calculator with a LOG key.
- Always begin by isolating the base and the power on one side of the equation.
- Once isolated, rewrite the equation in logarithmic form. (see separate lesson)
- Apply the change of base property to rewrite and put in the calculator as a common log.
- If the missing variable was not the only part of the exponent, some final Algebra may be required to find the final value for x.
Video Highlights
- 00:00 Introduction
- 00:22 \(3(2.4)^x=30\) example of solving for a missing exponent
- 03:47 \(40+30(2.1)^{x-1}=100\) example of solving for a missing variable in the exponent
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