Algebra 2
Exponential and logarithmic functions
Logarithms (definition and evaluation)
20 practice questions
0 video lessons
Theory + worked examples
Logarithms: Definition and Evaluation
Common Core Algebra 2 • Standard F-BF.5 • Exponential & Logarithmic Functions
Logarithms: Definition and Evaluation is a topic in Exponential & Logarithmic Functions in the Common Core State Standards. It is aligned to Standard F-BF.5, which requires students to understand logarithms as the inverse of exponential functions.
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:
\[\log_b x=y\quad\Longleftrightarrow\quad b^y=x.\]
- \(\log x\) means base 10.
- \(\ln x\) means base \(e\).
- The graph is the reflection of \(b^x\) over \(y=x\).
Read it as a question: “\(b\) to what power gives \(x\)?”
\(\log_2 x\) is the inverse of \(2^x\).
The definition of a logarithm.
The definition:
\[\log_b x=y\ \Longleftrightarrow\ b^y=x\]
\(\log_b b=1\) and \(\log_b 1=0\) for any base.
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.
Example 1 — Evaluate a log
Evaluate \(\log_2 8\).
Solution
Ask: \(2\) to what power is \(8\)?
| \(2^3\) | \(=\) | \(8\) |
| \(\log_2 8\) | \(=\) | \(3\) |
Example 2 — Base 10
Evaluate \(\log 1000\).
Solution
Base 10: \(10^3=1000\).
| \(\log 1000\) | \(=\) | \(3\) |
Example 3 — A negative result
Evaluate \(\log_5 \dfrac{1}{25}\).
Solution
\(5^{-2}=\dfrac1{25}\).
| \(\log_5\dfrac1{25}\) | \(=\) | \(-2\) |
Example 4 — Exponential to log form
Rewrite \(3^4=81\) in logarithmic form.
Solution
Base stays the base; exponent is the log.
| \(3^4=81\) | \(\Rightarrow\) | \(\log_3 81=4\) |
Common pitfalls
A log is an exponent — that's the answer you seek.
You can't take the log of \(0\) or a negative number.
\(\log x\) is base 10, \(\ln x\) is base \(e\).
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.
More in Exponential and logarithmic functions