Common and Natural Logarithms

In this lesson, we will look into common logarithms and natural logarithms.

Common Logarithms

Logarithms to base 10 are called common logarithms. We often write “log₁₀” as “log” or “lg”. Common logarithms can be evaluated using a scientific calculator.

Recall that by the definition of logarithm.

log Y = X ↔ Y = 10^X

Natural Logarithms

Besides base 10, another important base is e. Log to base e are called natural (or Naperian) logarithms.. “loge” are often abbreviated as “ln”. Natural logarithms can also be evaluated using a scientific calculator.

By definition

ln Y = X ↔ Y = e^X

Using a calculator, we can use common and natural logarithms to solve equations of the form a^x = b, especially when b cannot be expressed as aⁿ.

Example:

Solve the equations

a) 6^{x + 2} = 21
b) e^2x = 9

Solution:

a) 6^{x + 2} = 21
log 6^{x + 2} = log 21
(x + 2) log 6 = log 21

b) e^2x = 9
ln e^2x = ln 9
3x ln e = ln 9
3x = ln 9

Example:

Express 3^x(2^2x) = 7(5^x) in the form a^x = b. Hence, find x.

Solution:

the equation becomes

12^x = 7(5^x)

Videos

Common and Natural Logarithms
We can use many bases for a logarithm, but the bases most typically used are the bases of the common logarithm and the natural logarithm. The common logarithm has base 10, and is represented on the calculator as log(x). The natural logarithm has base e, a famous irrational number, and is represented on the calculator by ln(x). The natural and common logarithm can be found throughout Algebra and Calculus.

Common and Natural Logarithms.
Properties of Logarithms

Custom Search