Logarithmic models
Logarithmic Functions
Logarithmic Functions is a topic in Exponential & Logarithmic Functions in the California Common Core State Standards. It is aligned to Standard F-BF.5, which requires students to understand logarithms as the inverses of exponential functions.
A logarithmic function \(\log_b x\) is the inverse of \(b^x\), answering “what exponent gives \(x\)?” and defined only for \(x>0\).
Theory
A logarithm answers “what exponent?” It is the inverse of an exponential function:
Because it inverts \(b^x\), the log graph is the reflection of the exponential across \(y=x\): its domain is \(x>0\), and it has a vertical asymptote at \(x=0\).
Two logs get special names: the common log \(\log=\log_{10}\) and the natural log \(\ln=\log_e\).
The definition and its inverse relationships:
How to work with logarithms
- To evaluate: rewrite as “base to what power gives the input?”
- To convert: \(\log_b x=y\leftrightarrow b^y=x\).
- For the domain: set the inside \(>0\) and solve.
Ask: 2 to what power is 32?
| \(2^5\) | \(=\) | \(32\) |
| \(\log_2 32\) | \(=\) | \(5\) |
\(\log_b x=y\) means \(b^y=x\).
| \(\log_3 81=4\) | \(\Longleftrightarrow\) | \(3^4=81\) |
\(\ln\) is \(\log_e\), and logs undo the matching exponential.
| \(\ln e^3\) | \(=\) | \(3\) |
The input of a logarithm must be positive.
| \(x-5\) | \(>\) | \(0\) |
| \(x\) | \(>\) | \(5\) |
Domain: \(x>5\), or \((5,\infty)\).
Common pitfalls
Frequently asked questions
What is a logarithm?
The inverse of an exponential: \(\log_b x=y\) means \(b^y=x\). It returns the exponent needed to reach \(x\).
What is the difference between log and ln?
\(\log\) is base 10 (the common log); \(\ln\) is base \(e\) (the natural log).
What is the domain of a logarithmic function?
Its input must be positive, so the domain of \(\log_b x\) is \(x>0\).
How are logs and exponentials related?
They are inverses: \(\log_b(b^x)=x\) and \(b^{\log_b x}=x\), so each undoes the other.