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

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### Common Logarithms

### Natural Logarithms

^{x}(2^{2x}) = 3^{x}(2^{2})^{x} = (3 × 4)^{x} = 12^{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} = Plog_{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.

1. log x = log2x^{2} - 2

2. ln x + ln (x + 1) = 5

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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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

**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

3*x* ln e = ln 9

3*x* = ln 9

**Example:**

Express 3* ^{x}*(2

**Solution:**

the equation becomes

12* ^{x }*= 7(5

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.

Example: Write the following logarithms in exponential form. Evaluate if possible.

The logarithm of a Product:

log

The logarithm of a Quotient:

log

The logarithm of a number raised to a power:

log

How to use the properties of logarithms to condense and solve logarithms?

How to use the properties of logarithms to expand logarithms?

Examples:

Solve without a calculator:

log

log 1

log

ln e

Solve with a calculator:

log 3

log 32

ln √5

ln 7.3

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.

1. log x = log2x

2. ln x + ln (x + 1) = 5

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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