🟣 Logarithmic Functions
Unlock the inverse of exponentials! Master logarithms, their properties, and applications in science and engineering.
🔍 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.
Definition
The logarithm of a number $x$ with base $b$ is written as:
$$\log_b(x) = y$$
This is equivalent to:
$$b^y = x$$
"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 $e^1=e$
- $\log_5(125)=3$ because $5^3=125$
Special Logarithms
- 📌 Common Logarithm: If no base is written, the default base is 10 (i.e., $\log(x)$ means $\log_{10}(x)$)
- 📌 Natural Logarithm: Written as $\ln(x)$, has a base of $e$ (approximately 2.718)
⚙️ Properties of Logarithms
Logarithms follow important rules that make calculations easier:
1️⃣ Product Rule
$$\log_b(MN)=\log_b M+\log_b N$$
Example: $\log_2(8 \cdot 4) = \log_2(8) + \log_2(4) = 3 + 2 = 5$
2️⃣ Quotient Rule
$$\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N$$
Example: $\log_{10}(100/10) = \log_{10}(100) - \log_{10}(10) = 2 - 1 = 1$
3️⃣ Power Rule
$$\log_b(M^p)=p\log_b M$$
Example: $\log_2(8^2) = 2\log_2(8) = 2(3) = 6$
4️⃣ Change of Base Formula
$$\log_b M=\frac{\log_k M}{\log_k b}$$
Example: $\log_2(8)=\frac{\ln(8)}{\ln(2)} = \frac{2.079}{0.693} = 3$
This is useful for calculating logs on calculators that only have $\log$ and $\ln$ buttons!
5️⃣ Special Values
- $\log_b(1) = 0$ for any base $b$ (because $b^0 = 1$)
- $\log_b(b) = 1$ for any base $b$ (because $b^1 = b$)
- $\log_b(b^x) = x$ (inverse property)
- $b^{\log_b(x)} = x$ (inverse property)
✏️ Solving Logarithmic Equations
To solve equations involving logarithms, rewrite them in exponential form.
Example 1: Basic Equation
Solve $\log_2(x)=3$
Rewrite as: $x=2^3=8$
Answer: $x=8$
Example 2: With Addition
Solve $\log_{10}(x+1)=2$
Rewrite as: $x+1=10^2=100$
Therefore: $x=99$
Answer: $x=99$
Example 3: With Coefficient
Solve $2\log_3(x)=4$
Divide by 2: $\log_3(x)=2$
Then: $x=3^2=9$
Answer: $x=9$
Example 4: Using Properties
Solve $\log_2(x) + \log_2(x-3) = 2$
Use product rule: $\log_2(x(x-3)) = 2$
Rewrite: $x(x-3) = 2^2 = 4$
Expand: $x^2 - 3x = 4$
Solve: $x^2 - 3x - 4 = 0$, so $(x-4)(x+1)=0$
Since $x$ must be positive (domain restriction), Answer: $x=4$
📉 Logarithmic Graphs
The graph of $y=\log_b(x)$ is the inverse of $y=b^x$. It has distinctive characteristics:
Key Graph Features
- A vertical asymptote at $x=0$
- The function is undefined for $x \leq 0$
- Domain: $(0, \infty)$
- Range: $(-\infty, \infty)$
- X-intercept at $(1, 0)$ (since $\log_b(1) = 0$)
- If $b>1$, the graph increases; if $0
- Graph passes through $(b, 1)$ since $\log_b(b) = 1$
Example: Graphing $y=\log_2(x)$
| $x$ | $y$ |
|---|---|
| 0.5 | -1 |
| 1 | 0 |
| 2 | 1 |
| 4 | 2 |
| 8 | 3 |
The graph passes through these points and rises slowly to the right.
🌍 Real-Life Applications of Logarithms
1️⃣ Earthquakes (Richter Scale)
Magnitude is measured on a logarithmic scale
Formula: $M = \log_{10}\left(\frac{I}{I_0}\right)$
Example: If one earthquake is 1000 times stronger than another:
$\log_{10}(1000)=3$, so the difference in magnitude is 3 units
2️⃣ pH Scale (Chemistry)
pH is defined as: $\text{pH}=-\log[H^+]$
Example: If hydrogen ion concentration is $1\times10^{-5}$:
$\text{pH} = -\log(10^{-5}) = 5$
3️⃣ Sound Intensity (Decibels)
Loudness formula: $\text{dB}=10\log_{10}(I/I_0)$
Example: If one sound is 1000 times more intense:
$10\log_{10}(1000) = 10(3) = 30$ dB louder
4️⃣ Radioactive Decay & Half-Life
To find half-life $t_{1/2}$ from decay equation $N(t) = N_0 e^{-kt}$:
Set $N(t) = \frac{N_0}{2}$ and solve using logarithms:
$t_{1/2} = \frac{\ln(2)}{k}$
5️⃣ Information Theory (Computer Science)
Number of bits needed to represent $n$ items:
$\text{bits} = \lceil \log_2(n) \rceil$
🎯 Practice Questions
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🔥 Challenge Questions
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