Logarithm Definition: Exponents and Logarithms - Mathematics (Grade 10)

Understanding Logarithms

Logarithm is a mathematical operation that is the inverse of exponentiation. If we have an exponential equation $b = a^c$ , then its logarithmic form is $^a\log b = c$ .

Formal Definition of Logarithms

Let $a$ be a positive number where $0 < a < 1$ or $a > 1$ , and $b > 0$ , then:

^a\log b = c \text{ if and only if } b = a^c

Where:

$a$ is the base of the logarithm
$b$ is the number whose logarithm we are finding (numerus)
$c$ is the result of the logarithm

We can read $^a\log b = c$ as: $a$ raised to what power equals $b$ ? The answer is $c$ . Because $a^c = b$ .

Relationship Between Exponents and Logarithms

Logarithms and exponents are related as operations that are inverses of each other. Consider the following examples:

Exponential Form	Logarithmic Form
$2^5 = 32$	$^2\log 32 = 5$
$3^2 = 9$	$^3\log 9 = 2$
$5^{-2} = \frac{1}{25}$	$^5\log \frac{1}{25} = -2$
$7^0 = 1$	$^7\log 1 = 0$

Common Logarithm (Base 10)

Logarithm with base 10 is called the common logarithm. It is often simplified by omitting the base 10:

^{10}\log a = \log a

Applications of Logarithms in Exponential Growth

Determining Time to Reach a Specific Quantity

A bacterial colony initially consists of 2,000 bacteria that divide every 1 hour. The growth of these bacteria follows an exponential function:

f(x) = 2,000(2^x)

where $x$ is time in hours.

Then, to determine the time needed for bacteria to reach a specific number, for example 100,000 bacteria, we need to find the value of $x$ that satisfies:

100,000 = 2,000(2^x)

By dividing both sides by 2,000:

50 = 2^x

To find the value of $x$ , we use the concept of logarithms:

x = \log_2 50

This demonstrates that logarithms are very useful tools for solving exponential equations, especially when finding the exponent value that yields a specific result.