Logarithms (definition and evaluation)
Logarithms: Definition and Evaluation
Logarithms: Definition and Evaluation is a topic in Exponential & Logarithmic Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.5(C), which requires students to rewrite exponential equations as their corresponding logarithmic equations and vice versa.
A logarithm answers “what exponent gives this number?” — \(\log_b x=y\) means \(b^y=x\), the inverse of the exponential.
Theory
A logarithm is the inverse of an exponential — it returns the exponent:
- \(\log x\) means base 10.
- \(\ln x\) means base \(e\).
- The graph is the reflection of \(b^x\) over \(y=x\).
The definition:
How to evaluate a log
- Write it as \(b^{?}=x\).
- Find the exponent that works.
- That exponent is the value of the log.
- Convert to exponential form when in doubt.
Ask: \(2\) to what power is \(8\)?
| \(2^3\) | \(=\) | \(8\) |
| \(\log_2 8\) | \(=\) | \(3\) |
Base 10: \(10^3=1000\).
| \(\log 1000\) | \(=\) | \(3\) |
\(5^{-2}=\dfrac1{25}\).
| \(\log_5\dfrac1{25}\) | \(=\) | \(-2\) |
Base stays the base; exponent is the log.
| \(3^4=81\) | \(\Rightarrow\) | \(\log_3 81=4\) |
Common pitfalls
Frequently asked questions
What is a logarithm?
The inverse of an exponential; it gives the exponent.
What does \(\log_2 8=3\) mean?
\(2^3=8\).
What is the difference between \(\log\) and \(\ln\)?
\(\log\) is base 10; \(\ln\) is base \(e\).
Can you take the log of a negative number?
No — the argument must be positive.