1. Introduction
Logarithmic identities are special formulas that help simplify expressions quickly. These identities come directly from the basic definition of logarithms and show how logs behave in certain standard cases.
They are different from log laws because they deal with specific values, not general operations.
2. Identity: \(\log_b 1 = 0\)
The logarithm of 1 to any valid base is always 0.
Reason: \( b^0 = 1 \), so the exponent needed is 0.
2.1. Examples
\( \log_{10} 1 = 0 \)
\( \log_3 1 = 0 \)
\( \log_{2} 1 = 0 \)
3. Identity: \(\log_b b = 1\)
The logarithm of a base to itself is always 1.
Reason: \( b^1 = b \)
3.1. Examples
\( \log_{5} 5 = 1 \)
\( \log_{2} 2 = 1 \)
\( \log_{10} 10 = 1 \)
4. Identity: \( \log_b (b^x) = x \)
This identity shows that logarithms and exponents undo each other.
If you take a log of a base raised to a power, the result is that power.
4.1. Examples
\( \log_2 (2^5) = 5 \)
\( \log_3 (3^4) = 4 \)
\( \log_{10} (10^6) = 6 \)
5. Identity: \( b^{\log_b x} = x \)
This is the reverse of the previous identity. The exponent cancels the logarithm, giving back the original number.
5.1. Examples
\( 2^{\log_2 7} = 7 \)
\( 10^{\log_{10} 3} = 3 \)
\( 5^{\log_5 12} = 12 \)
6. Identity: Log of Reciprocal
\( \log_b \left( \dfrac{1}{x} \right) = -\log_b x \)
Reason: \( \dfrac{1}{x} = x^{-1} \), and using the power rule gives:
\( \log_b (x^{-1}) = -\log_b x \)
6.1. Examples
\( \log_{10} (1/100) = -\log_{10} 100 = -2 \)
\( \log_2 (1/8) = -3 \)
7. Identity: Log With Power Base
\( \log_{b^n} x = \dfrac{1}{n} \log_b x \)
This identity helps when the base is a power itself.
7.1. Examples
\( \log_{4} 8 = \log_{2^2} 8 = \dfrac{1}{2} \log_2 8 = \dfrac{1}{2} (3) = 1.5 \)
\( \log_{9} 3 = \log_{3^2} 3 = \dfrac{1}{2} \log_3 3 = \dfrac{1}{2} (1) = 0.5 \)
8. Examples
- \( \log_{5} 1 = 0 \)
- \( \log_6 (6^3) = 3 \)
- \( 7^{\log_7 2} = 2 \)
- \( \log_4 (1/16) = -2 \)
9. Common Mistakes
- Forgetting that logs are defined only for positive values.
- Confusing law of logs with identities.
- Incorrectly applying log rules to addition inside the log.
- Thinking \( \log_b b = 0 \) (correct value is 1).
10. Quick Practice
Simplify using identities:
- \( \log_7 1 \)
- \( \log_9 (9^4) \)
- \( 5^{\log_5 11} \)
- \( \log_{3^3} 27 \)
11. Summary
- Logs and exponents are inverse operations.
- \( \log_b 1 = 0 \) and \( \log_b b = 1 \).
- \( \log_b (b^x) = x \) and \( b^{\log_b x} = x \).
- Reciprocal log gives a negative value.
- Power bases change the value by a factor of \( 1/n \).