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.

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= loga x if and only if x = ay

Then the function is given by

f(x) = loga 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 log10 or simply log.

f(x) = log10 x

Natural Logarithmic Function

The logarithmic function to the base e is called the natural logarithmic function and it is denoted by loge.

f(x) = loge 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

logb MN = logb M + logb N

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

Example : log 30 + log 2 = log 60
Quotient Rule 

logb M/N = logb M – logb N

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

Example : log8 56 – log8 7 = log8(56/7)=log88 = 1

Power Rule 

Raise an exponential expression to power and multiply the exponents.

Logb Mp = P logb M

Example : log 1003 = 3. Log 100 = 3 x 2 = 6

Zero Exponent Rule

loga 1 = 0.

Change of Base Rule

logb (x) = ln x / ln b or logb (x) = log10 x / log10 b

Other Important Rules of Logarithmic Function

  • Logb b = 1 Example : log1010 = 1
  • Logb bx = x Example : log1010x = x
  • \(b^{\log _{b}x}=x\) . Substitute y= logbx , it becomes by = 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(ax) = 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, loga b = y becomes ay = b

(4/9)y = 27/8

(22/32)y = 33 / 23

(⅔)2y = (3/2)3

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 log9 x + 7 log9 y – 3 log9 z

Solution:

By using the power rule , Logb Mp = P logb M, we can write the given equation as

5 log9 x + 7 log9 y – 3 log9 z = log9 x5 + log9 y7 – log9 z3

From product rule, logb MN = logb M + logb N

5 log9 x + 7 log9 y – 3 log9 z = log9 x5y7 – log9 z3

From Quotient rule, logb M/N = logb M – logb N

5 log9 x + 7 log9 y – 3 log9 z = log9 (x5y7 / z3 )

Therefore, the single logarithm is 5 log9 x + 7 log9 y – 3 log9 z = log9 (x5y7 / z3 )

Question 2:

Use the properties of logarithms to write as a single logarithm for the given equation: 1/2 log2 x – 8 log2 y – 5 log2 z

Solution:

By using the power rule , Logb Mp = P logb M, we can write the given equation as

1/2 log2 x – 8 log2 y – 5 log2 z = log2 x1/2 – log2 y8 – log2 z5

From product rule, logb MN = logb M + logb N

Take minus ‘- ‘ as common

1/2 log2 x – 8 log2 y – 5 log2 z = log2 x1/2 – log2 y8z5

From Quotient rule, logb M/N = logb M – logb N

1/2 log2 x – 8 log2 y – 5 log2 z = log2 (x1/2 / y8z5 )

The solution is

1/2 log2 x – 8 log2 y – 5 log2 z = \(\log _{2}\left ( \frac{\sqrt{x}}{y^{8}z^{5}} \right )\)

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