Exponential Equations Worksheet 1: Practice & Solutions
Table of Contents :
Exponential equations are an essential component of algebra and higher-level mathematics. They involve variables in the exponent, creating equations that can model a variety of real-world situations, from finance to population growth. In this article, we'll delve into the practice and solutions surrounding exponential equations, giving you the tools to master this important mathematical concept.
Understanding Exponential Equations
Exponential equations take the form:
[ a^x = b ]
Where:
- ( a ) is the base,
- ( x ) is the exponent (the variable),
- ( b ) is a constant.
Key Properties of Exponents
To solve exponential equations effectively, it's crucial to understand some fundamental properties of exponents:
- Product of Powers: ( a^m \cdot a^n = a^{m+n} )
- Quotient of Powers: ( \frac{a^m}{a^n} = a^{m-n} )
- Power of a Power: ( (a^m)^n = a^{m \cdot n} )
- Zero Exponent: ( a^0 = 1 ) (provided ( a \neq 0 ))
- Negative Exponent: ( a^{-n} = \frac{1}{a^n} ) (provided ( a \neq 0 ))
Practice Problems
Here are some practice problems on exponential equations. Try solving them on your own before looking at the solutions.
Problem Set
- Solve for ( x ): ( 2^x = 16 )
- Solve for ( x ): ( 3^{x-1} = 27 )
- Solve for ( x ): ( 5^{2x} = 125 )
- Solve for ( x ): ( 4^{x+1} = 64 )
- Solve for ( x ): ( 10^{3x} = 1000 )
Table of Problems and Solutions
Here’s a table of the problems alongside their solutions:
<table> <tr> <th>Problem</th> <th>Solution</th> </tr> <tr> <td>1. ( 2^x = 16 )</td> <td>( x = 4 )</td> </tr> <tr> <td>2. ( 3^{x-1} = 27 )</td> <td>( x = 4 )</td> </tr> <tr> <td>3. ( 5^{2x} = 125 )</td> <td>( x = 1.5 )</td> </tr> <tr> <td>4. ( 4^{x+1} = 64 )</td> <td>( x = 2 )</td> </tr> <tr> <td>5. ( 10^{3x} = 1000 )</td> <td>( x = 1 )</td> </tr> </table>
Solutions Breakdown
Let's go through each problem and break down the solutions step by step.
Problem 1: ( 2^x = 16 )
To solve this, we can rewrite 16 as ( 2^4 ):
[ 2^x = 2^4 ]
By equating the exponents, we have:
[ x = 4 ]
Problem 2: ( 3^{x-1} = 27 )
Recognize that 27 can be rewritten as ( 3^3 ):
[ 3^{x-1} = 3^3 ]
This leads us to:
[ x - 1 = 3 ]
Thus,
[ x = 4 ]
Problem 3: ( 5^{2x} = 125 )
Rewriting 125 as ( 5^3 ):
[ 5^{2x} = 5^3 ]
Setting the exponents equal gives:
[ 2x = 3 ]
Solving for ( x ):
[ x = \frac{3}{2} = 1.5 ]
Problem 4: ( 4^{x+1} = 64 )
Since ( 64 = 4^3 ):
[ 4^{x+1} = 4^3 ]
We conclude:
[ x + 1 = 3 ]
Thus,
[ x = 2 ]
Problem 5: ( 10^{3x} = 1000 )
Noticing that ( 1000 = 10^3 ):
[ 10^{3x} = 10^3 ]
This leads to:
[ 3x = 3 ]
So,
[ x = 1 ]
Conclusion
Mastering exponential equations requires practice and a solid understanding of the properties of exponents. By working through problems and using the solutions provided, you can improve your skills and confidence in solving these types of equations. Remember, practice makes perfect! 🚀 Keep working on more problems, and soon you'll find exponential equations to be a piece of cake! 🍰