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

**Related Pages**

Logarithmic Functions

Properties Or Rules Of Logarithms

Rules Of Exponents

Logarithm Rules

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

Logarithms to base 10 are called **common logarithms**. We often write
“log_{10}” 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}

Besides base 10, another important base is e. Log to base e are called **natural
logarithms**. “log_{e}” 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 a^{n}.

**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^{3x} = 9

ln e^{3x} = 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:**

Since 3^{x}(2^{2x}) = 3^{x}(2^{2})^{x} = (3 × 4)^{x} = 12^{x}
the equation becomes

12^{x }= 7(5^{x})

**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:

log_{b}MN = log_{b}M + log_{b}N

The logarithm of a Quotient:

log_{b}M/N = log_{b}M - log_{b}N

The logarithm of a number raised to a power:

log_{b}M^{P} = P log_{b}M

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**

**Examples:**

Solve without a calculator:

log_{3}3

log 1

log_{16}2

ln e^{3}

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.

**Examples:**

Solve, round to four decimal places.

- log x = log2x
^{2}- 2 - ln x + ln (x + 1) = 5

Try the free Mathway calculator and
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