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

Evaluating logarithmic functions using a calculator - log and Ln Professor Edward Burger explains evaluating logarithmic functions using a calculator.

Common and Natural Logarithms.
Properties of Logarithms

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