FUNCTIONS 09
Logarithmic Functions
A logarithm answers an exponent question: which power of a base produces a given positive number?
The main idea
For \(b>0\) and \(b\ne1\),
\[\boxed{\log_b x=y\quad\Longleftrightarrow\quad b^y=x}.\]The logarithm is therefore the inverse operation of exponentiation.
1. Reading a logarithm
Since
\[2^3=8,\]we can write
\[\log_2 8=3.\]So \(\log_2 8\) means “the power of \(2\) that gives \(8\)”.
⇕
\(\log_2 8=3\)
same relationship, read in reverse
2. Exponentials and logarithms are inverses
The functions \(y=e^x\) and \(y=\ln x\) undo one another.
Their graphs are mirror images across the line \(y=x\).
3. Domain of a logarithm
For real logarithms, the input must be positive:
\[\boxed{x>0}.\]The curve \(y=\ln x\) approaches the vertical line \(x=0\) but never reaches or crosses it.
4. Important points on the graph
Every logarithmic function satisfies
\[\log_b1=0,\qquad \log_b b=1.\]So \(y=\log_bx\) always passes through \((1,0)\) and \((b,1)\).
5. The base controls whether the graph rises or falls
If \(b>1\), \(\log_bx\) is increasing. If \(0
6. Logarithm laws
For positive \(x\) and \(y\),
\[\log_b(xy)=\log_bx+\log_by,\]\[\log_b\!\left(\frac{x}{y}\right)=\log_bx-\log_by,\]\[\log_b(x^r)=r\log_bx.\]These laws turn multiplication into addition, division into subtraction and powers into multiplication.
quotient → difference
power → multiplier
7. The natural logarithm
The logarithm with base \(e\) is written
\[\boxed{\ln x=\log_e x}.\]It is especially important in calculus because it is the inverse of \(e^x\).
⇕
\(y=\ln x\)
8. Inverse identities
Because \(e^x\) and \(\ln x\) undo one another,
\[\boxed{\ln(e^x)=x}\]for every real \(x\), and
\[\boxed{e^{\ln x}=x}\]for \(x>0\).
\(x\xrightarrow{\ln}\ln x\xrightarrow{e^{(\cdot)}}x\)
9. Solving exponential equations
Logarithms allow the unknown exponent to be brought down.
If
\[e^{2x}=7,\]then
\[2x=\ln7,\qquad x=\frac{\ln7}{2}.\]↓ take \(\ln\)
\(2x=\ln7\)
↓
\(x=\ln7/2\)
10. Why \(\ln x\) matters for calculus
The natural logarithm has the derivative
\[\boxed{\frac{d}{dx}\ln x=\frac1x},\qquad x>0.\]This connects logarithmic functions directly to rational functions and later to integration.
↓ differentiate
\(1/x\)
logarithms ↔ rational functions
A logarithm is an inverse exponent. Its real domain is \(x>0\), its graph has vertical asymptote \(x=0\), and \(\ln x\) is central to calculus because it is the inverse of \(e^x\) and has derivative \(1/x\).