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Common and Natural Logarithms

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In this lesson, we will learn common logarithms and natural logarithms and how to solve problems using common log and natural log.

The following diagrams gives the definition of Logarithm, Common Log, and Natural Log. Scroll down the page for more examples and solutions.

Definition of Logarithm

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 = 10X

Natural Logarithms

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

By definition

ln Y = X ↔ Y = eX

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


Solve the equations
a) 6x + 2 = 21
b) e2x = 9


a) 6x + 2 = 21
log 6x + 2 = log 21
(x + 2) log 6 = log 21

b) e3x = 9
ln e3x = ln 9
3x ln e = ln 9
3x = ln 9


Express 3x(22x) = 7(5x) in the form ax = b. Hence, find x.


Since 3x(22x) = 3x(22)x = (3 × 4)x = 12x

the equation becomes

12x = 7(5x)

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.

Defines common log, log x, and natural log, ln x, and works through examples and problems using a calculator. Common and Natural Logarithms
Example: Write the following logarithms in exponential form. Evaluate if possible.
Properties of Logarithms
The logarithm of a Product:
logbMN = logbM + logbN
The logarithm of a Quotient:
logbM/N = logbM - logbN
The logarithm of a number raised to a power:
logbMP = P logbM
How to use the properties of logarithms to condense and solve logarithms?
How to use the properties of logarithms to expand logarithms? Common and Natural Logs
Solve without a calculator:
log 1
ln e3

Solve with a calculator:
log 3
log 32
ln √5
ln 7.3 How to solve logarithmic equations?
The first example is with common logs and the second example is natural logs. It is good to remember the properties of logarithms also can be applied to natural logs.
Solve, round to four decimal places.
1. log x = log2x2 - 2
2. ln x + ln (x + 1) = 5

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