CK-12-Calculus

http://www.ck12.org Chapter 6. Transcendental Functions

This is equivalent to the exponential form

by=x.

For example, the following table shows the logarithmic forms in the first row and the corresponding exponential
forms in the second row.

Logarithmic Form→ log 216 = 4 log 5251 =− 2 log 10100 = 2 logee= 1

Exponential Form→ 24 = 16 5 −^2 = 251 102 = 100 e^1 =e

Historically, logarithms with base of 10 were very popular. They are called the common logarithms. Recently the
base 2 has been gaining popularity due to its considerable role in the field of computer science and the associated
binary number system. However, the most widely used base in applications is the natural logarithm, which has
an irrational base denoted bye,in honor of the famous mathematician Leonhard Euler. This irrational constant is
e≈ 2. 718281 .Formally, it is defined as the limit of( 1 +x)^1 /xasxapproaches zero. That is,

limx→ 0 ( 1 +x)^1 /x=e.

We denote the natural logarithm ofxby lnxrather than logex.So keep in mind, that lnxis the power to whichemust
be raised to producex.That is, the following two expressions are equivalent:

y=lnx⇐⇒x=ey

The table below shows this operation.

Natural Logarithm ln ln 2= 0. 693 ln 1= 0 lne= 1 lne^3 = 3

Equivalent Exponential Form e^0.^693 = 2 e^0 = 1 e^1 =e e^3 =e^3

A Comparison between Logarithmic Functions and Exponential Functions
Looking at the two graphs of exponential functions above, we notice that both pass the horizontal line test. This
means that an exponential function is a one-to-one function and thus has an inverse. To find a formula for this
inverse, we start with the exponential function

y=bx.

Interchangingxandy,

x=by.

Projecting the logarithm to the basebon both sides,