Solving for Missing Variables in the Exponents When the Bases Are the Same or Can Be Made the Same
Description/Explanation/Highlights
Video Description
This video describes how to solve for a missing exponent when the bases are the same or can be made the same.
Steps and Key Points to Remember
To solve exponential equations with the same base or bases that can be made the same, follow these steps:
- When the bases are the same on both sides of the equal sign that implies that for the equation to be equal
the exponents must also be equal i.e. In the example: \(4^{2x-3}=4^{x+3} \) since both of the bases have the same vaue of 4, then \(2x-3\) must equal \(x+3\) - Set the exponents equal to each other i.e. In the above example: \(2x-3=x+3\)
- This creates an equation that should be solved by the appropriate method. In the above example, it is a linear equation and should be solved like any other linear equation. Subtract x from both sides and
add 3 to both sides. x=6 - To check your answer, plug the answer you got in for each occurence of x in the original problem i.e. \(4^{2(6)-3}=4^{(6)+3} \)
- When the bases are NOT the same on both sides of the equal sign we can often make them the same if they are multiples of the same number, either one of the bases or a smaller number i.e. In the example: \(4^{x-1}=8^{3x+4} \), 4 and 8 are both multiples of 2 so we can take advantage of this.
- Re-write the bases substituting the new base and power for the original base in the problem i.e. \(4=2^2\) and \(8=2^3 \)
- Substitute back into the problem for the original bases i.e. In this example: \({(2^2)}^{x-1}={(2^3)}^{3x+4} \)
- Use the rules of exponents to simplify. In this example multiply the exponents to raise a power to a power. Don’t forget to distribute as needed. It is helpful to use parenthesis. i.e. In this example: \({(2^2)}^{x-1}={(2^3)}^{3x+4}\Longrightarrow 2^{2x-2}=2^{9x+12}\)
- Once the bases are the same, set the exponents equal to each other and solve as before. \( 2x-2=9x+12\)
- Fractions can often be re-written as negative exponents to get the bases the same. i.e. \(\frac{1}{16}=2^{-4} \) with a base of 2.
Here are some key points to keep in mind when solving for missing variables in exponential equations:
- The bases must be the same or be made the same before the exponents can be set equal to each other.
- Once the exponents are set equal to each other, use the method appropriate to solve the resulting equation. For example, if the resulting equation is linear, solve as a linear equation by getting the variable on one side and the numbers on the other.
- Answers can always be checked by plugging the result into the variables on the original problem and solving both sides to be sure the sides are equal.
- When changing bases to get them equal, see if one of the bases works as a multiple of the other. If so, you may need to change only one of the bases.
- If neither base can be used as the new base, look for a number that is smaller than both numbers and is a multiple of both.
- When a new base (and power) is substituted for the original base, be sure to enclose it in parenthesis and put the original power on the outside.
- Don’t forget to distribute when raising a new exponent to the original power.
- Fractions can often be re-written to an integer base with a negative exponent to create like bases.
Video Highlights
- 00:00 Introduction to solving exponential equations
- 00:12 \(4^{2x-3}=4^{x+3} \)example of solving when the bases are the same.
- 02:00 \(4^{x-1}=8^{3x+4} \) example of solving when the bases are different
- 04:18 \(\frac{1}{16}=8^{2x+2} \) example of solving using negative exponents when one base is a fraction
- 06:10 Conclusion