Lesson 8 Exponents and Exponential Functions portfolio
In this lesson, we learned about exponents and exponential functions. Exponents are a way to represent repeated multiplication of a number by itself, while exponential functions are functions where the variable is in the exponent.
For this portfolio, I will demonstrate my understanding of exponents and exponential functions through various examples and problems.
1. Evaluate the following expressions:
a) \(2^3\)
b) \(5^4\)
a) \(2^3 = 2 x 2 x 2 = 8\)
b) \(5^4 = 5 x 5 x 5 x 5 = 625\)
2. Solve the following exponential equations:
a) \(3^x = 81\)
b) \(2^x = 16\)
a) To solve \(3^x = 81\), we can rewrite 81 as \(3^4\). So, \(x = 4\).
b) To solve \(2^x = 16\), we can rewrite 16 as \(2^4\). So, \(x = 4\).
3. Graph the following exponential functions:
a) \(f(x) = 2^x\)
b) \(g(x) = 3^x\)
a) For \(f(x) = 2^x\), the graph will be a curve that starts at the point (0,1) and increases rapidly as x increases.
b) For \(g(x) = 3^x\), the graph will be a curve that starts at the point (0,1) and increases even more rapidly than \(f(x)\) as x increases.
4. Identify the base and exponent in the following exponential expressions:
a) \(4^{3x}\)
b) \(e^x\)
a) For \(4^{3x}\), the base is 4 and the exponent is \(3x\).
b) For \(e^x\), the base is the constant e (approximately 2.71828) and the exponent is x.
5. Solve the following exponential growth/decay problems:
a) A population of bacteria doubles every hour. If there are 100 bacteria initially, how many will there be after 5 hours?
b) A radioactive substance decays by 20% every hour. If there are 500 grams of the substance initially, how much will remain after 3 hours?
a) Since the population doubles every hour, the growth rate is 2. Therefore, after 5 hours, there will be \(100 x 2^5 = 3200\) bacteria.
b) Since the substance decays by 20% every hour, the decay rate is 0.8. Therefore, after 3 hours, there will be \(500 x 0.8^3 \approx 256\) grams of the substance remaining.
Overall, this portfolio demonstrates my understanding of exponents and exponential functions through various examples and problems.