Logarithms

In day to day to life, we often sometimes say that something grows exponentially. Say we invest some money somewhere, it grows exponentially every year. Now growing exponentially can mean money became double, triple etc

For example,  $\blueD2$start color blueD, 2, end color blueD raised to the $\greenE4^\text{th}$start color greenE, 4, end color greenE, start superscript, t, h, end superscript power equals $\goldD{16}$start color goldD, 16, end color goldD. This is expressed by the exponential equation $\blueD2^\greenE4=\goldD{16}$start color blueD, 2, end color blueD, start superscript, start color greenE, 4, end color greenE, end superscript, equals, start color goldD, 16, end color goldD.

2*2*2*2 = 16

What if the opposite happens. Opposite of exponential growth is logarithmic growth.

In plain terms, if we keep multiplying something with X times will be exponential and if we keep dividing something X times will be logarithmic.

24=16⟺log2(16)=4

Both equations describe the same relationship between the numbers $2$start color blueD, 2, end color blueD, $4$start color greenE, 4, end color greenE, and $16$start color goldD, 16, end color goldD, where $2$start color blueD, 2, end color blueD is the base, $4$start color greenE, 4, end color greenE is the exponent, and $16$start color goldD, 16, end color goldD is the power. 

The difference is that while the exponential form isolates the power, $16$start color goldD, 16, end color goldD, the logarithmic form isolates the exponent, $4$start color greenD, 4, end color greenD.

Here are more examples of equivalent logarithmic and exponential equations. 

Logarithmic form		Exponential form
${\log }_2\left(8\right)=3$log, start subscript, start color blueD, 2, end color blueD, end subscript, left parenthesis, start color goldD, 8, end color goldD, right parenthesis, equals, start color greenD, 3, end color greenD	⟺	$2^3=8$start color blueD, 2, end color blueD, start superscript, start color greenD, 3, end color greenD, end superscript, equals, start color goldD, 8, end color goldD
${\log }_3\left(81\right)=4$log, start subscript, start color blueD, 3, end color blueD, end subscript, left parenthesis, start color goldD, 81, end color goldD, right parenthesis, equals, start color greenD, 4, end color greenD	⟺	$3^4=81$start color blueD, 3, end color blueD, start superscript, start color greenD, 4, end color greenD, end superscript, equals, start color goldD, 81, end color goldD
${\log }_5\left(25\right)=2$log, start subscript, start color blueD, 5, end color blueD, end subscript, left parenthesis, start color goldD, 25, end color goldD, right parenthesis, equals, start color greenD, 2, end color greenD	⟺	$5^2=25$start color blueD, 5, end color blueD, start superscript, start color greenD, 2, end color greenD, end superscript, equals, start color goldD, 25, end color goldD

Definition of a logarithm

Generalizing the examples above leads us to the formal definition of a logarithm. 

logb(a)=c⟺bc=a

Both equations describe the same relationship between  $a$start color goldD, a, end color goldD, $b$start color blueD, b, end color blueD, and $c$start color greenE, c, end color greenE:

• $b$start color blueD, b, end color blueD is the $$start color blueD, b, a, s, e, end color blueD,
• $c$start color greenE, c, end color greenE is the $$start color greenE, e, x, p, o, n, e, n, t, end color greenE, and
• $a$start color goldD, a, end color goldD is the $$start color goldD, p, o, w, e, r, end color goldD.

log, start subscript, start color blueD, 2, end color blueD, end subscript, left parenthesis, start color goldD, 16, end color goldD, right parenthesis, equals, start color greenE, 4, end color greenE