Logarithm & Antilogarithm — Terms, Rules, and Examples
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Core Definitions
- Logarithm: If \(b^x=N\) with \(b>0,\; b\neq 1,\; N>0\), then \[ \log_b(N)=x. \] It is “the power to which the base \(b\) must be raised to obtain \(N\)”.
- Common logarithm: Base \(10\): \(\log_{10}N\) or \(\log N\).
- Natural logarithm: Base \(e\;( \approx 2.71828)\): \(\ln N=\log_e N\).
- Characteristic & Mantissa (base 10): For \(\log_{10}N\),
the characteristic is the integer part and the mantissa is the fractional part.
Example ( \(N>1\) ): \[ \log_{10}(250)=2.3979 \quad\Rightarrow\quad \text{Characteristic}=2,\;\text{Mantissa}=0.3979. \]Example ( \(0<N<1\) ): \[ \log_{10}(0.0045) = -3 + 0.6532 \;=\; \overline{3}.6532, \] where the bar on \(3\) denotes a negative characteristic with positive mantissa.
- Antilogarithm: The inverse of a logarithm. If \(x=\log_b(N)\) then \[ \operatorname{antilog}_b(x)=N=b^x. \]
Fundamental Rules of Logarithms
1) Product Rule
\[ \log_b(MN)=\log_b M + \log_b N \]
Log of a product equals the sum of logs.
2) Quotient Rule
\[ \log_b\!\left(\frac{M}{N}\right)=\log_b M – \log_b N \]
Log of a quotient equals the difference of logs.
3) Power Rule
\[ \log_b(M^k)=k\,\log_b M \]
Exponent becomes a multiplier.
4) Root Rule
\[ \log_b\!\big(\sqrt[n]{M}\big)=\frac{1}{n}\,\log_b M \]
An \(n\)-th root is a power of \(\tfrac{1}{n}\).
5) Log of 1 & Base
\[ \log_b(1)=0,\qquad \log_b(b)=1. \]
6) Change of Base
\[ \log_b(M)=\frac{\log_k(M)}{\log_k(b)} \quad (\text{often } k=10 \text{ or } e). \]
Domains: \(b>0,\; b\neq 1,\; M>0,\; N>0\).
Worked Examples
A. Using Product, Quotient, and Power Rules
Example A1: \(\log_{10}(2000)\)
\[ \log_{10}(2\times 10^3)=\log_{10}2 + \log_{10}(10^3) = \log_{10}2 + 3 \approx 0.3010 + 3 = 3.3010. \]
Example A2: \(\log_{10}\!\left(\dfrac{50}{2}\right)\)
\[ \log_{10}50 – \log_{10}2 \approx 1.6990 – 0.3010 = 1.3980. \]
Example A3: \(\log_2(32)\)
\[ \log_2(2^5) = 5\,\log_2 2 = 5. \]
Example A4: \(\log_{10}\!\big(\sqrt[3]{1000}\big)\)
\[ \frac{1}{3}\log_{10}(1000)=\frac{1}{3}\cdot 3=1. \]
B. Change of Base
\[ \log_{3}(20)=\frac{\ln(20)}{\ln(3)} \approx \frac{2.9957}{1.0986}\approx 2.728. \]
C. Antilogarithms (Base 10)
Example C1: If \(\log_{10}(N)=2.3010\), then \[ N = \operatorname{antilog}_{10}(2.3010)=10^{2.3010}\approx 200. \]
Example C2: If \(\log_{10}(N)=\overline{1}.4771\) (i.e., \(-1+0.4771\)), then \[ N = 10^{-1+0.4771}=10^{-1}\cdot 10^{0.4771}\approx 0.1\times 3=0.3. \]
Quick Reference
| Quantity | Formula / Value | Note |
|---|---|---|
| Definition | \(b^x=N \iff \log_b N = x\) | \(b>0,\; b\neq 1,\; N>0\) |
| Product | \(\log_b(MN)=\log_b M+\log_b N\) | Sum of logs |
| Quotient | \(\log_b\!\left(\dfrac{M}{N}\right)=\log_b M-\log_b N\) | Difference of logs |
| Power | \(\log_b(M^k)=k\,\log_b M\) | Exponent to multiplier |
| Root | \(\log_b(\sqrt[n]{M})=\tfrac{1}{n}\log_b M\) | \(n\in\mathbb{N}\) |
| Change of Base | \(\log_b M=\dfrac{\log_k M}{\log_k b}\) | Use \(k=10\) or \(k=e\) |
| Antilog | \(\operatorname{antilog}_b(x)=b^x\) | Inverse of \(\log_b\) |