# Logarithmic Functions

In Mathematics, before the discovery of calculus, many Math scholars used logarithms to change multiplication and division problems into addition and subtraction problems. In Logarithms, the power is raised to some numbers (usually, base number) to get some other number. It is an inverse function of exponential function. We know that Mathematics and Science constantly deal with the large powers of numbers, logarithms are most important and useful. In this article, we are going to discuss the definition and formula for the logarithmic function, rules and properties, examples in detail.

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## Logarithmic Function Definition

In mathematics, the logarithmic function is an inverse function to exponentiation. The logarithmic function is defined as

For x > 0 , a > 0, and a\(\neq\)1,

y= log_{a }x if and only if x = a^{y}

Then the function is given by

**f(x) = log _{a }x**

The base of the logarithm is a. This can be read it as log base a of x. The most 2 common bases used in logarithmic functions are base 10 and base e.

* Also, try out:* Logarithm Calculator

### Common Logarithmic Function

The logarithmic function with base 10 is called the common logarithmic function and it is denoted by log_{10} or simply log.

f(x) = log_{10} x

### Natural Logarithmic Function

The logarithmic function to the base e is called the natural logarithmic function and it is denoted by log_{e}^{.}

f(x) = log_{e} x

## Logarithmic Functions Properties

Logarithmic Functions have some of the properties that allow you to simplify the logarithms when the input is in the form of product, quotient or the value taken to the power. Some of the properties are listed below.

**Product Rule**

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

Multiply two numbers with the same base, then add the exponents.

Example : log 30 + log 2 = log 60

**Quotient Rule **

log_{b }M/N = log_{b} M – log_{b }N

Divide two numbers with the same base, subtract the exponents.

Example : log_{8} 56 – log_{8} 7 = log_{8}(56/7)=log_{8}8 = 1

**Power Rule **

Raise an exponential expression to power and multiply the exponents.

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

Example : log 100^{3} = 3. Log 100 = 3 x 2 = 6

**Zero Exponent Rule **

log_{a} 1 = 0.

**Change of Base Rule**

log_{b }(x) = ln x / ln b or log_{b }(x) = log_{10} x / log_{10} b

**Other Important Rules of Logarithmic Function**

- Log
_{b }b = 1 Example : log_{10}10 = 1 - Log
_{b }b^{x }= x Example : log_{10}10^{x}= x - \(b^{\log _{b}x}=x\) . Substitute y= log
_{b}x , it becomes b^{y}= x

There are also some of the logarithmic function with fractions. It has a useful property to find the log of a fraction by applying the identities

- ln(ab)= ln(a)+ln(b)
- ln(a
^{x}) = x ln (a)

We also can have logarithmic function with fractional base.

Consider an example,

\(3\log _{\frac{4}{9}}\sqrt[4]{\frac{27}{8}}=\frac{3}{4}\log _{\frac{4}{9}}\frac{27}{8}\)By the definition, log_{a} b = y becomes a^{y} = b

(4/9)^{y} = 27/8

(2^{2}/3^{2})^{y} = 3^{3 }/ 2^{3}

(⅔)^{2y} = (3/2)^{3}

## Video Lesson

### Logarithmic Equations

## Logarithmic Function Examples

Here you are provided with some logarithmic functions example.

**Example 1:**

Use the properties of logarithms to write as a single logarithm for the given equation: 5 log_{9} x + 7 log_{9} y – 3 log_{9} z

**Solution:**

By using the power rule , Log_{b }M^{p} = P log_{b} M, we can write the given equation as

5 log_{9} x + 7 log_{9} y – 3 log_{9} z = log_{9} x^{5} + log_{9} y^{7} – log_{9} z^{3}

From product rule, log_{b }MN = log_{b} M + log_{b }N

5 log_{9} x + 7 log_{9} y – 3 log_{9} z = log_{9} x^{5}y^{7} – log_{9} z^{3}

From Quotient rule, log_{b }M/N = log_{b} M – log_{b }N

5 log_{9} x + 7 log_{9} y – 3 log_{9} z = log_{9} (x^{5}y^{7 } / z^{3} )

Therefore, the single logarithm is 5 log_{9} x + 7 log_{9} y – 3 log_{9} z = log_{9} (x^{5}y^{7 } / z^{3} )

**Question 2:**

Use the properties of logarithms to write as a single logarithm for the given equation: 1/2 log_{2} x – 8 log_{2} y – 5 log_{2} z

**Solution:**

By using the power rule , Log_{b }M^{p} = P log_{b} M, we can write the given equation as

1/2 log_{2} x – 8 log_{2} y – 5 log_{2} z = log_{2} x^{1/2} – log_{2} y^{8} – log_{2} z^{5}

From product rule, log_{b }MN = log_{b} M + log_{b }N

Take minus ‘- ‘ as common

1/2 log_{2} x – 8 log_{2} y – 5 log_{2} z = log_{2} x^{1/2} – log_{2} y^{8}z^{5}

From Quotient rule, log_{b }M/N = log_{b} M – log_{b }N

1/2 log_{2} x – 8 log_{2} y – 5 log_{2} z = log_{2} (x^{1/2} / y^{8}z^{5} )

The solution is

1/2 log_{2} x – 8 log_{2} y – 5 log_{2} z = \(\log _{2}\left ( \frac{\sqrt{x}}{y^{8}z^{5}} \right )\)

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