Definition of logarithms

Definition of Logarithms

A logarithm is the inverse operation to exponentiation. In simpler terms, a logarithm answers the question: to what exponent must a base be raised to produce a given number? For example, if you have the equation b^y = x, then the logarithm of x with base b is y. This can be written as log_b(x) = y.

Understanding Logarithmic Notation

The notation log_b(x) has three parts:

• b: the base of the logarithm. This must be a positive number, and it cannot be equal to 1.
• x: the argument of the logarithm. This must be a positive number.
• y: the logarithm itself, which represents the exponent to which the base must be raised to obtain x.

Common Logarithms

There are two special types of logarithms that you should know:

• Common logarithm: This is a logarithm with base 10. It is often written as log(x) without the base. For example, log(100) = 2 because 10² = 100.
• Natural logarithm: This is a logarithm with base e, where e is approximately equal to 2.71828. It is denoted as ln(x). For example, ln(e) = 1 because e¹ = e.

Examples of Logarithmic Definitions

Let us look at a few examples to understand how to define logarithms:

Example 1

Find log₂(8).

We need to find the exponent to which the base 2 must be raised to get 8.

We know that:

23 = 8

Therefore, log₂(8) = 3.

Example 2

Find log₁₀(1000).

We need to find the exponent to which the base 10 must be raised to get 1000.

We know that:

103 = 1000

Therefore, log₁₀(1000) = 3.

Example 3

Find ln(e⁴).

We need to find the exponent to which the base e must be raised to get e⁴.

Since:

e4 = e4

It follows that ln(e⁴) = 4.

Properties of Logarithms

Logarithms have several important properties that can help simplify calculations and solve equations. Here are some key properties:

• Product property: log_b(xy) = log_b(x) + log_b(y)
• Quotient property: log_b(x/y) = log_b(x) - log_b(y)
• Power property: log_b(x^k) = k × log_b(x)

Example of Using Logarithmic Properties

Let us apply the properties of logarithms in an example:

Example 4

Simplify log₃(27) + log₃(9).

Using the product property:

log₃(27) + log₃(9) = log₃(27 × 9)

Next, calculate 27 × 9:

27 × 9 = 243

Now we have:

log₃(243).

We know that:

35 = 243

Therefore, log₃(243) = 5.

Common Mistakes

Logarithmic Equations

Logarithmic equations can be solved using the definition of logarithms and the properties mentioned above. For example, consider the equation:

Example 5

Solve the equation log₅(x) = 2.

Using the definition of logarithms, we rewrite the equation in exponential form:

x = 52

Calculating the right side gives:

x = 25

Thus, the solution is x = 25.

Changing the Base of a Logarithm

Sometimes, it is useful to change the base of a logarithm. The change of base formula is:

log_b(x) = log_k(x) / log_k(b),

where k is any positive number except 1. This can help when using calculators that only have common or natural logarithm functions.

Example 6

Use the change of base formula to evaluate log₂(16).

Using base 10:

log2(16) = log(16) / log(2)

Using a calculator:

log(16) ≈ 1.2041

log(2) ≈ 0.3010

Now calculate:

log2(16) ≈ 1.2041 / 0.3010 ≈ 4

Thus, log₂(16) = 4.

Summary

• A logarithm is the inverse of exponentiation.
• The notation log_b(x) represents the exponent to which base b must be raised to yield x.
• Common and natural logarithms are special cases of logarithms.
• Logarithmic properties simplify calculations and solve equations.
• Use the change of base formula when necessary.

Check your understanding

1. What is the value of log₁₀(100)?
2. Using the properties of logarithms, simplify log₄(16) - log₄(4).
3. What is the result of ln(e³)?
4. Change the base of log₃(81) to base 9 and calculate the value.