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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 bx for b > 1:

As an example, first graph y = 2x and obtain the graph of y = log2x.

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)?


Example 2 Use the approximation log100.5 ≈ −0.301 to estimate log1020.

Example 3 Use the approximation log23 ≈ 1.585 and log25 ≈ 2.322 to estimate log245.

Example 4 Suppose A and b are positive numbers with log3A = 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

Richter value

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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