Logarithmic Functions
1️⃣ Understanding Logarithms
A logarithm is the inverse operation of an exponential function. Instead of multiplying repeatedly (as in exponentials), logarithms tell us how many times we multiply a base to reach a given number.
- ✔ The logarithm of a number \(x\) with base \(b\) is written as: \(\log_b(x)\)
- ✔ This is equivalent to: \(b^y = x\) where \(y=\log_b(x)\).
- ✔ “Logarithm = exponent” – it asks, “To what power must we raise \(b\) to get \(x\)?”
Examples:
- \(\log_2(8)=3\) because \(2^3=8\).
- \(\log_{10}(100)=2\) because \(10^2=100\).
- \(\ln(e)=1\) because \(e^1=e\).
📌 If no base is written, the default base is 10 (i.e., \(\log(x)\) means \(\log_{10}(x)\)).
📌 The natural logarithm, written as \(\ln(x)\), has a base of \(e\) (approximately 2.718).
2️⃣ Properties of Logarithms
Logarithms follow important rules that make calculations easier:
- Product Rule:
\[
\log_b(MN)=\log_b M+\log_b N
\]
Example: \(\log_2(8\cdot4)=3+2=5\). - Quotient Rule:
\[
\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N
\]
Example: \(\log_{10}(20/4)=\log_{10}(20)-\log_{10}(4)\). - Power Rule:
\[
\log_b(M^p)=p\log_b M
\]
Example: \(\log_2(8^2)=2\log_2(8)=6\). - Change of Base Formula:
\[
\log_b M=\frac{\log_k M}{\log_k b}
\]
Example: \(\log_2(8)=\frac{\ln8}{\ln2}\).
3️⃣ Solving Logarithmic Equations
To solve equations involving logarithms, rewrite them in exponential form.
Example 1: Solve \(\log_2(x)=3\).
Rewrite as: \(x=2^3=8\).
Answer: \(x=8\).
Example 2: Solve \(\log_{10}(x+1)=2\).
Rewrite as: \(x+1=10^2=100\), so \(x=99\).
Answer: \(x=99\).
Example 3: Solve \(2\log_3(x)=4\).
Divide by 2: \(\log_3(x)=2\); then \(x=3^2=9\).
Answer: \(x=9\).
4️⃣ Logarithmic Graphs
The graph of \(y=\log_b(x)\) is the inverse of \(y=b^x\). It has:
- A vertical asymptote at \(x=0\).
- The function is undefined for \(x\le0\).
- If \(b>1\), the graph increases; if \(0
Example: For \(y=\log_2(x)\):
| \(x\) | \(y\) |
|---|---|
| 1 | 0 |
| 2 | 1 |
| 4 | 2 |
| 8 | 3 |
The graph passes through these points and rises slowly to the right.
5️⃣ Real-Life Applications of Logarithms
- Earthquakes (Richter Scale): Magnitude is measured on a logarithmic scale.
- pH Scale (Chemistry): pH is defined as \( -\log[H^+] \).
- Sound Intensity (Decibels): \( \text{dB}=10\log_{10}(I/I_0) \) measures loudness.
- Radioactive Decay (Half-Life): Logarithms model decay over time.
Example (Earthquake Magnitude): If one earthquake is 1000 times stronger than another, then:
\(\log_{10}(1000)=3\)
The difference in magnitude is 3 units.
🎯 Practice Questions – Fundamental
- Convert the exponential form \(2^3=8\) to logarithmic form.
- Convert the logarithmic form \(\log_{10}(x)=2\) to exponential form.
- Solve for \(x\): \(\log_2(x)=4\).
- Solve for \(x\): \(\log_{10}(x+1)=3\).
- Use logarithm properties to simplify: \(\log_2(8\cdot4)\).
- Solve for \(x\): \(2\log_3(x)=6\).
- Find the value of \(x\) if \(f(x)=5\log(x)\) for \(x=2\) (assume base 10).
- Use the change of base formula to compute \(\log_2(16)\).
- Graph \(y=\log_2(x)\) and label key points.
- Find the domain and range of \(f(x)=\log_3(x-2)\).
🔥 Challenging Problems – Push Your Limits!
- Solve for \(x\): \(\log_2 (x^2-4)=3\).
- A scientist studying bacteria finds the population follows \(P(t)=500e^{0.2t}\). How long will it take for the population to reach 2000?
- The loudness of a sound (in decibels) is given by \(L=10\log\left(\frac{I}{I_0}\right)\). If one sound is 1000 times more intense than another, how much louder is it?
- Find the equation of a logarithmic function that passes through the points \( (1,0) \) and \( (4,2) \).
- The pH of a solution is given by \(\text{pH}=-\log[H^+]\). If the hydrogen ion concentration is \(1\times10^{-5}\), find the pH.
7️⃣ Summary
- ✅ Logarithms are the inverse of exponential functions.
- ✅ Key properties include the product, quotient, power rules, and change of base formula.
- ✅ Solving logarithmic equations requires rewriting them in exponential form.
- ✅ Logarithms model real-world phenomena such as earthquakes, pH, sound intensity, and radioactive decay.
8️⃣ Fantastic Job!
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