Test Your Knowledge: Real-World Logarithm & Exponent Applications

📚 Quick Study Guide

• 🌱 Exponential Growth: Understand that exponential growth is modeled by the formula $y = a(1+r)^t$, where $a$ is the initial amount, $r$ is the growth rate, and $t$ is the time.
• decay: Exponential decay is modeled by the formula $y = a(1-r)^t$, where $a$ is the initial amount, $r$ is the decay rate, and $t$ is the time.
• 🏦 Compound Interest: The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal, $r$ is the interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the time in years.
• 🌡️ Logarithms for Solving Exponentials: Use logarithms to solve for variables in the exponent. For example, if $2^x = 7$, then $x = \frac{\log(7)}{\log(2)}$.
• 🔊 Decibel Scale: The decibel scale for sound intensity is logarithmic, given by $dB = 10 \log_{10}(\frac{I}{I_0})$, where $I$ is the sound intensity and $I_0$ is the reference intensity.
• 📅 Half-Life: The half-life of a substance undergoing radioactive decay is the time it takes for half of the substance to decay. The formula is $N(t) = N_0(\frac{1}{2})^{\frac{t}{T}}$, where $N(t)$ is the remaining amount, $N_0$ is the initial amount, $t$ is the time, and $T$ is the half-life.
• 🌍 Richter Scale:** The Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale. An increase of 1 on the Richter scale represents a tenfold increase in amplitude.

🧪 Practice Quiz

1. A bacteria culture doubles every 3 hours. If you start with 100 bacteria, how many will there be after 9 hours?
   1. A. 300
   2. B. 600
   3. C. 800
   4. D. 1600
2. The half-life of a radioactive isotope is 10 years. If you start with 200 grams, how much will remain after 30 years?
   1. A. 25 grams
   2. B. 50 grams
   3. C. 75 grams
   4. D. 100 grams
3. An investment of $5000 earns 5% interest compounded annually. How much will the investment be worth after 5 years?
   1. A. $6288.95
   2. B. $6381.41
   3. C. $6000.00
   4. D. $5625.00
4. The population of a town grows at a rate of 3% per year. If the initial population is 10,000, what will the population be in 10 years?
   1. A. 12,000
   2. B. 13,439
   3. C. 14,000
   4. D. 15,000
5. The intensity of a sound is measured as $10^{-6}$ W/m². What is the decibel level of this sound (given $I_0 = 10^{-12}$ W/m²)?
   1. A. 30 dB
   2. B. 60 dB
   3. C. 90 dB
   4. D. 120 dB
6. The magnitude of an earthquake is measured as 6 on the Richter scale. How many times greater is the amplitude of this earthquake compared to an earthquake with a magnitude of 4?
   1. A. 2 times
   2. B. 10 times
   3. C. 100 times
   4. D. 1000 times
7. A car depreciates at a rate of 15% per year. If the car was initially purchased for $25,000, what will its value be after 3 years?
   1. A. $12,914.38
   2. B. $15,343.75
   3. C. $16,281.25
   4. D. $17,000.00