# Logarithmic Functions

GOAL: Learn logarithmic functions as inverses of exponential functions and
use them to model various

interesting situations, like intensity of earthquake, noise level, and acidity
of beer.

Q1: What “undoes” the exponential function (e.g. If then )

A1: The logarithmic function with base b, denoted
(If
)

Definition:
is defined by

**Example 1** Express the following logarithms as an integer or fraction
without using a calculator.

•The graph of log _{b}x for b > 1:

As an example, first graph y = 2^{x} and obtain the graph of y = log_{2}x.

Properties of logarithmic functions

• It’s continuous and increasing.

Note: The most common choices for b are **10, e and 2 .**

• The laws of logarithms . (Reversing the laws of exponents) Let s, t > 0. Then

Q2: Can you explain property (1)?

A2:

**Example 2** Use the approximation log_{10}0.5 ≈
−0.301 to estimate log_{10}20.

**Example 3** Use the approximation log_{2}3 ≈
1.585 and log_{2}5 ≈ 2.322 to estimate log_{2}45.

**Example 4** Suppose A and b are positive numbers with log_{3}A =
b. Write
in
terms of b .

**Example 5** A bank teller claims that a saving account with principal of
$1000 earning interest at a

annual rate of 1.3 %, compounded weekly , after T years would at least double.
What is the smallest

possible T in whole years?

**• Logarithms with base 10**

Logarithms with base 10, called common logarithms , are used in many well-known applications.

**1 The Richter scale**

where A is the amplitude of the seismic wave of a reference earthquake and x
is the amplitude of the

seismic wave of the earthquake in question.

**Example 6** One of the worst earthquakes in history occured in Tokyo and
registered 8.3 on the Richter

scale. A more recent earthquake in California in 1989 registered 7.2. How much
more severe was the

earthquake in Tokyo in terms of the amplitude of its seismic wave?

**2 The decibel scale**

Noise level in decibels

where I is the amplitude of a minimal audible sound wave and x is the
amplitude of another sound

wave. Read Text Example 2.3.3 (Pg 141).

**3 The pH scale**

pH value =

where [H^{+}] is the concentration of hydrogen ions in a solution. Read Text
Example 2.3.4 (Pg 142).

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