4.3: Logarithmic Functions

This section studies the inverse of $f(x) = a^x$.

Recall the graph of $a^x$:

The exponential function $f(x) = a^x$ has two possible graph shapes:
$a > 1$ $0 < a < 1$
$a > 1$ $0 < a < 1$

$a^x$ is one-to-one because it passes the horizontal line test.

An inverse of $a^x$ must exist! We call it the logarithm.

Logarithmic Functions

logarithmic function
Suppose $a > 0, a \neq 1$. The logarithmic function with base $a$ is defined by \[\log_a x = y \qquad \text{means} \qquad a^y = x\]

Both equations have names: \[\underbrace{\log_a x = y}_{\Large\text{logarithmic form}} \qquad \qquad \underbrace{a^y = x}_{\Large{\text{exponential form}}}\]

Writing $\log_a x$ means you are plugging in $x$ into $\log_a$.
You can use parentheses for readability: $\log_a(x)$

To find logs, convert to exponential form and find what power $a$ needs to be taken to in order to get $x$.

Evaluate the following:
  1. $\log_28 $
  2. $\log_{10}100$
  3. $\log_{16}4$
  4. $\log_2\left(\frac{1}{2}\right)$

Few useful facts for logarithms:

Properties of Logarithms
  1. $\log_a 1 = 0$
  2. $\log_a a = 1$
  3. $\log_a a^x = x$
  4. $a^{\log_a x} = x$
Evaluate the following:
  1. $\log_51 $
  2. $\log_{5}5$
  3. $\log_{4}4^8$
  4. $6^{\log_6 12}$

Here are two logarithms which will be extensively used.

common logarithm
Base 10 logarithm, the base is omitted: \[\log x = \log_{10} x\]
natural logarithm
Base $e$ logarithm, denoted by $\ln$: \[\ln x = \log_{e} x\]

We can rewrite the properties of logarithms for the natural logarithm:

Properties of Natural Logarithms
  1. $\ln 1 = 0$
  2. $\ln e = 1$
  3. $\ln e^x = x$
  4. $e^{\ln x} = x$
Evaluate:
  1. $\ln e^8$
  2. $\ln \left(\frac{1}{e^2}\right)$

Logarithmic Graphs

Let's inspect what the graph looks like.

The function $f(x) = \log_a x$ has the general shape In general:
Sketch a graph of $f(x) = 1 + \log_{10}(x - 3)$.

The Decibel scale is a function between intensity (energy required) and perceived loudness by a human.

For example, a normal conversation is around 60 dB while an electric vacuum is around 80 dB. But how much more energy was used to generate a difference of 20 dB?

Decibels $B$ for a sound with intensity $I$ is calculated by \[B = 10\log\left(\dfrac{I}{I_0}\right)\] where $I_0$ is the intensity of complete silence (0 dB).
Find the decibel level of a sound whose intensity $I$ is 100 times of $I_0$.

It takes 100 times more energy to generate a difference of 20 dB!