Solve The Equation: ${ 2\left(4^{2x+1}\right) = 128 }$
Introduction
Solving equations involving exponents can be a challenging task, especially when dealing with variables in the exponent. In this article, we will focus on solving the equation 2(4^(2x+1)) = 128, which involves a variable in the exponent. We will use various mathematical techniques to simplify the equation and solve for the variable x.
Understanding the Equation
The given equation is 2(4^(2x+1)) = 128. To start solving this equation, we need to understand the properties of exponents and how to simplify expressions involving exponents.
Properties of Exponents
Exponents are a shorthand way of writing repeated multiplication. For example, 4^2 can be written as 4 Γ 4, which equals 16. When dealing with exponents, we need to remember the following properties:
- Product of Powers: When multiplying two powers with the same base, we add the exponents. For example, 4^2 Γ 4^3 = 4^(2+3) = 4^5.
- Power of a Power: When raising a power to another power, we multiply the exponents. For example, (42)3 = 4^(2Γ3) = 4^6.
- Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, 4^0 = 1.
Simplifying the Equation
To simplify the equation 2(4^(2x+1)) = 128, we can start by using the property of exponents that states 4^2 = 16. We can rewrite 4^(2x+1) as (42)(x+1/2), which equals 16^(x+1/2).
import math
# define the equation
def equation(x):
return 2 * (16 ** (x + 1/2)) - 128
Using Logarithms to Solve the Equation
One way to solve the equation is to use logarithms. We can take the logarithm of both sides of the equation to get rid of the exponent.
Logarithmic Properties
When dealing with logarithms, we need to remember the following properties:
- Logarithm of a Product: The logarithm of a product is equal to the sum of the logarithms. For example, log(4 Γ 5) = log(4) + log(5).
- Logarithm of a Power: The logarithm of a power is equal to the exponent multiplied by the logarithm of the base. For example, log(4^2) = 2 Γ log(4).
Applying Logarithmic Properties
We can apply the logarithmic properties to the equation 2(4^(2x+1)) = 128. Taking the logarithm of both sides, we get:
log(2(4^(2x+1))) = log(128)
Using the logarithmic property of a product, we can rewrite the left-hand side as:
log(2) + log(4^(2x+1))
Using the logarithmic property of a power, we can rewrite the right-hand side as:
log(2) + (2x+1) Γ log(4)
Solving for x
Now we have an equation in the form of:
log(2) + (2x+1) Γ log(4) = log(128)
We can simplify the equation by using the fact that log(128) = log(2^7) = 7 Γ log(2).
import math
# define the equation
def equation(x):
return log(2) + (2*x + 1) * log(4) - 7 * log(2)
Isolating x
To isolate x, we can subtract log(2) from both sides of the equation:
(2x+1) Γ log(4) = 6 Γ log(2)
We can then divide both sides by log(4):
2x+1 = 6 Γ log(2) / log(4)
Simplifying the Equation
We can simplify the equation by using the fact that log(4) = log(2^2) = 2 Γ log(2).
import math
# define the equation
def equation(x):
return 2*x + 1 - 6 * log(2) / (2 * log(2))
Solving for x
Now we have a simple linear equation in the form of:
2x+1 = 3
We can solve for x by subtracting 1 from both sides:
2x = 2
Dividing both sides by 2, we get:
x = 1
Conclusion
In this article, we solved the equation 2(4^(2x+1)) = 128 using various mathematical techniques, including logarithms and exponent properties. We started by simplifying the equation using exponent properties and then used logarithms to get rid of the exponent. Finally, we isolated x and solved for its value. The solution to the equation is x = 1.
References
- [1] "Exponents and Logarithms" by Math Open Reference
- [2] "Solving Equations with Exponents" by Khan Academy
- [3] "Logarithmic Properties" by Wolfram MathWorld
Introduction
In our previous article, we solved the equation 2(4^(2x+1)) = 128 using various mathematical techniques, including logarithms and exponent properties. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on solving equations involving exponents.
Q&A
Q: What is the main concept behind solving equations involving exponents?
A: The main concept behind solving equations involving exponents is to use logarithms to get rid of the exponent. This allows us to isolate the variable and solve for its value.
Q: How do I simplify expressions involving exponents?
A: To simplify expressions involving exponents, you can use the following properties:
- Product of Powers: When multiplying two powers with the same base, you add the exponents. For example, 4^2 Γ 4^3 = 4^(2+3) = 4^5.
- Power of a Power: When raising a power to another power, you multiply the exponents. For example, (42)3 = 4^(2Γ3) = 4^6.
- Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, 4^0 = 1.
Q: What is the difference between a logarithm and an exponent?
A: A logarithm is the inverse operation of an exponent. While an exponent raises a number to a power, a logarithm finds the power to which a number must be raised to produce a given value.
Q: How do I use logarithms to solve equations involving exponents?
A: To use logarithms to solve equations involving exponents, you can follow these steps:
- Take the logarithm of both sides of the equation.
- Use the logarithmic property of a product to rewrite the left-hand side.
- Use the logarithmic property of a power to rewrite the right-hand side.
- Simplify the equation and isolate the variable.
Q: What are some common mistakes to avoid when solving equations involving exponents?
A: Some common mistakes to avoid when solving equations involving exponents include:
- Not using logarithms: Failing to use logarithms can make it difficult to isolate the variable and solve for its value.
- Not simplifying expressions: Failing to simplify expressions involving exponents can make it difficult to isolate the variable and solve for its value.
- Not checking units: Failing to check units can lead to incorrect solutions.
Q: How do I check my work when solving equations involving exponents?
A: To check your work when solving equations involving exponents, you can follow these steps:
- Plug in the solution into the original equation.
- Simplify the equation and check if it is true.
- If the equation is not true, recheck your work and try again.
Conclusion
In this article, we provided a Q&A section to help clarify any doubts and provide additional information on solving equations involving exponents. We covered topics such as simplifying expressions, using logarithms, and checking work. By following these tips and avoiding common mistakes, you can become more confident and proficient in solving equations involving exponents.
References
- [1] "Exponents and Logarithms" by Math Open Reference
- [2] "Solving Equations with Exponents" by Khan Academy
- [3] "Logarithmic Properties" by Wolfram MathWorld