Practical Uses of Logarithms: Examples with Base 10 and Base e

📚 Quick Study Guide

🔢 Logarithms are the inverse operation to exponentiation. If $b^y = x$, then $\log_b(x) = y$. 🧪 Common logarithms are logarithms with base 10, denoted as $\log_{10}(x)$ or simply $\log(x)$. 🌿 Natural logarithms are logarithms with base $e$ (Euler's number, approximately 2.71828), denoted as $\log_e(x)$ or $\ln(x)$. 🌍 Logarithms are used in various fields like: measuring earthquake intensity (Richter scale), sound intensity (decibels), and acidity (pH scale). 📈 Logarithmic scales are useful for representing data that spans a large range of values. 💡 Key properties: $\log(AB) = \log(A) + \log(B)$, $\log(\frac{A}{B}) = \log(A) - \log(B)$, and $\log(A^n) = n\log(A)$.

Practice Quiz

1. What is the approximate value of $\log(1000)$?
1. 0
2. 1
3. 3
4. 10
2. The Richter scale uses base-10 logarithms to measure the magnitude of earthquakes. How much more intense is an earthquake with a magnitude of 6 compared to one with a magnitude of 4?
1. 2 times
2. 20 times
3. 100 times
4. 1000 times
3. What is the value of $\ln(e^5)$?
1. 0
2. 1
3. 5
4. $e$
4. The formula for compound interest is $A = P(1 + r)^t$, where A is the final amount, P is the principal, r is the interest rate, and t is the time. How can logarithms be used to solve for t?
1. $t = \frac{\log(A/P)}{\log(1+r)}$
2. $t = \frac{\log(1+r)}{\log(A/P)}$
3. $t = \log(A/P) - \log(1+r)$
4. $t = \log(A/P) + \log(1+r)$
5. What is the pH of a solution if $[H^+] = 1 \times 10^{-7}$? (pH = -log[H+])
1. -7
2. 0
3. 1
4. 7
6. Simplify: $\log(25) + \log(4)$
1. $\log(29)$
2. $\log(21)$
3. 2
4. 100
7. Which of the following is equivalent to $\ln(\frac{a}{b})$?
1. $\ln(a) + \ln(b)$
2. $\ln(a) - \ln(b)$
3. $\frac{\ln(a)}{\ln(b)}$
4. $\ln(a*b)$