Logarithmic models
Logarithmic Functions
Logarithmic Functions is a topic in Exponential & Logarithmic Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.2(N), P.5(H), which requires students to analyze situations modeled by logarithmic functions and solve logarithmic equations.
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.