1.5: Logarithms

Inverse Functions

A function $f$ is called one-to-one if it never takes on the same value twice. In symbols: $f(x_1) \neq f(x_2)$ whenever $x_1 \neq x_2$.
Suppose $f(1) = 2, f(2) = 1, f(3) = 2$.
  1. Is $f$ a function?
  2. Is $f$ one-to-one?

The graph can be used to quickly determine if a function is one-to-one.

Horizontal Line Test
A function is one-to-one if no horizontal line intersects the graph more than once.
Suppose $f(x) = x^2$.
  1. Is $f$ one-to-one?
  2. If it isn't, restrict the domain of $f$ so that it is one-to-one.
Suppose $f$ is a one-to-one function. Then its inverse function $f^{-1}$ is defined by \[f^{-1}(y) = x \text{ whenever } f(x) = y\]

Intuition: $f^{-1}$ acts in the opposite direction as $f$.

Why is one-to-one needed to define the inverse?

Suppose $f(1) = 2, f(2) = 1, f(3) = 2$. Why does $f^{-1}$ not exist?
The function $f(x) = x^3$ has inverse function $f^{-1}(x) = x^{1/3}$. Show $f^{-1}$ is actually the inverse using the definition.
Inverse Function Property Let $f$ be a one-to-one function with domain $A$ and range $B$. The inverse function $f^{-1}$ satisfies the following properties \begin{align} f^{-1}(f(x))=x \qquad &\text{ for every } x \text{ in } A\\ f(f^{-1}(x))=x \qquad &\text{ for every } x \text{ in } B \end{align}
Are the functions \[f(x) = 2 - 5x \qquad g(x) = \dfrac{x - 2}{5}\] inverses? Show using the inverse function property.

Logarithms

The exponential function $f(x) = b^x$ for $b > 0, b\neq 1$ is one-to-one. It's inverse is called the logarithm with base $b$.

If $f(x) = b^x$ with $b > 0, b\neq 1$, then the logarithm function with base $b$ is the function $g(x) = \log_b(x)$ where \[\log_bx = y \text{ whenever } b^y = x\]

Think of $\log_bx$ as the exponent you need to raise $b$ to in order to get $x$.

Find the following logarithms:
  1. $\log_28$
  2. $\log_{10}100$
  3. $\log_3\frac{1}{9}$

If you apply the inverse function property, you get the following equations:

\[\log_b(b^x) = x \qquad b^{\log_bx} = x\]

Moreover, the laws of exponents carry over to the logarithms as well:

Laws of Logarithms If $x, y$ are positive numbers, then
  1. $\log_b(xy) = \log_b x + \log_b y$
  2. $\log_b\left(\frac{x}{y}\right) = \log_bx - \log_by$
  3. $\log_b(x^r) = r\log_bx$ for any real number $r$
Simplify $\log_2 80 - \log_2 5$.
Combine the following into one logarithm:
  1. $\log_3 x + \log_3 y$
  2. $2{\log _4}x + 5{\log _4}y - \frac{1}{2}{\log _4}z$
  3. $3\ln \left( {t + 5} \right) - 4\ln t - 2\ln \left( {s - 1} \right)$

Natural Logarithms

Logarithms with base $e$ is denoted by $\ln$ and is called the natural logarithm.

Simplify the following:
  1. $\ln e$
  2. $\ln a + \dfrac{1}{2}\ln b$

To solve equations involving log and exponents, apply the opposite operation to both sides.

Solve the following equations:
  1. $\ln x = 5$
  2. $e^{5 - 3x} = 10$
  3. $\ln x + \ln (x - 1) = 1$