# Log Formulas

Before learning log formulas, let us recall what are logs (logarithms). A logarithm is just another way of writing exponents. When we cannot solve a problem using the exponents, then we use logarithms. There are different logarithm formulas that are derived by using the laws of exponents. Let us learn them using a few solved examples.

## What are Log Formulas?

Before going to learn the log formulas, let us recall a few things. There are two types of logarithms, common logarithm (which is written as "log" and its base is 10 if not mentioned) and natural logarithm (which is written as "ln" and its base is always "e"). The below logarithm formulas are shown for common logarithms. However, they are all applicable for natural logarithms as well. Here are the most commonly used **log formulas**.

- log
_{b}1 = 0 - log
_{b}b = 1 - log
_{b}(xy) = log_{b}x + log_{b}y - log
_{b}(x / y) = log_{b}x - log_{b}y - log
_{b}a^{x}= x log_{b}a - log
_{b}a = (log_{c}a) / (log_{c}b)

Some of these rules have specific names like log_{b} (xy) = log_{b} x + log_{b} y is called the product formula of logs. In the same way, all the properties along with their names are mentioned in the table below.

## Logarithmic Formulas Derivation

Here is the derivation of some important log formulas. We use the laws of exponents in the derivation of log formulas.

### Product Formula of logarithms

The product formula of logs is, log_{b} (xy) = log_{b} x + log_{b} y.

**Derivation:**

Let us assume that log_{b} x = m and log_{b} y = n. Then by the definition of logarithm,

x = b^{m} and y = b^{n}.

Then xy = b^{m} × b^{n} = b^{m + n} (by a law of exponents, a^{m }× a^{n} = a^{m + n})

Converting xy = b^{m + n} into logarithmic form, we get

m + n = log_{b} xy

Substituting the values log_{b} x = m and log_{b} y = n here,

log_{b} (xy) = log_{b} x + log_{b} y

### Quotient Formula of logarithms

The quotient formula of logs is, log_{b} (x/y) = log_{b} x - log_{b} y.

**Derivation:**

Let us assume that log_{b} x = m and log_{b} y = n. Then by the definition of logarithm,

x = b^{m} and y = b^{n}.

Then x/y = b^{m} / b^{n} = b^{m - n }(by a law of exponents, a^{m }/ a^{n} = a^{m - n})

Converting x/y = b^{m - n} into logarithmic form, we get

m - n = log_{b} (x/y)

Substituting the values log_{b} x = m and log_{b} y = n here,

log_{b} (x/y) = log_{b} x - log_{b} y

### Power Formula of Logarithms

The power formula of logarithms says log_{b} a^{x} = x log_{b} a.

**Derivation:**

Let log_{b} a = m. Then by the definition of logarithm, a = b^{m}.

Raising both sides by x, we get

a^{x} = (b^{m})^{x}

a^{x} = b^{mx} (by a law of exponents, (a^{m})^{n} = a^{mn})

Converting this back into logarithmic form,

log_{b} a^{x} = m x

Substitute m = log_{b} a here,

log_{b} a^{x} = x log_{b} a

### Change of Base Formula of Logarithms

The change of base formula of logs says log_{b} a = (log_{c} a) / (log_{c} b).

**Derivation:**

Assume that log_{b} a = x, log_{c} a = y, and log_{c} b = z.

Converting these into exponential forms,

a = b^{x} ... (1)

a = c^{y} ... (2)

b = c^{z} ... (3)

From (1) and (2),

b^{x} = c^{y}

(c^{z})^{x} = c^{y} (from (3))

c^{zx} = c^{y}

Since the bases are the same, the powers also should be the same.

zx = y (or) x = y / z.

Substituting the values of x, y, and z here back,

log_{b} a = (log_{c} a) / (log_{c} b).

## Examples Using Logarithm Formulas

**Example 1:** Convert the following from exponential form to logarithmic form using the log formulas. a) 5^{3} = 125 b) 3^{-3} = 1 / 27.

**Solution:**

Using the definition of the logarithm,

b^{x} = a ⇒ log_{b} a = x

Using this,

a) 5^{3} = 125 ⇒ log_{5} 125 = 3

b) 3^{-3} = 1 / 27 ⇒ log_{3} 1/27 = -3

**Answer**: a) log_{5} 125 = 3; b) log_{3} 1/27 = -3.

**Example 2:** Compress the following expression as a single logarithm by using logarithmic formulas. 5 log x + log y - 8 log z.

**Solution:**

To find: The compressed form of the given expression as a single logarithm using logarithm formulas.

5 log x + log y - 8 log z

= (5 log x - 8 log z) + log y (Regrouped the terms)

= (log x^{5} - log z^{8}) + log y (∵ a log x = log x^{a})

= log (x^{5}/z^{8}) + log y (∵ log x - log y = log (x/y) )

= log (x^{5}y/z^{8}) (∵ log x + log y = log (xy) )

**Answer**: 5 log x + log y - 8 log z = log (x^{5}y/z^{8}).

**Example 3:** Find the integer value of log_{3} (1/9) using log formulas.

**Solution:**

log_{3} (1/9) = log_{3} 1 - log_{3} 9 (∵ log_{b} (x / y) = log_{b} x - log_{b} y)

= 0 - log_{3} 3^{2} (∵ log_{b} 1 = 0)

= - 2 log_{3} 3 (∵ log_{b} a^{x} = x log_{b} a)

= -2 (1) (∵ log_{b} b = 1)

= -2

**Answer:** log_{3} (1/9) = -2.

## FAQs on Log Formulas

### What are Logarithm Formulas?

The **logarithm formulas** are related to logarithms and are very helpful while solving the problems of logarithms. Some important log formulas are:

- log
_{b}(xy) = log_{b}x + log_{b}y - log
_{b}(x / y) = log_{b}x - log_{b}y - log
_{b}a^{x}= x log_{b}a - log
_{b}a = (log_{c}a) / (log_{c}b)

### How To Derive Log Formulas?

The laws of exponents are used to derive the log formulas. We also use the definition of logarithm while deriving the log formulas. i.e. we convert the logarithmic form into exponential form and vice versa in the derivation. For a detailed derivation of log formulas, you can refer to the "What are Log Formulas?" section of this page.

### What are the Applications of Log Formulas?

The problems that cannot be solved using the exponents' properties can be solved using logs. The log formulas are used to either compress a group of logarithms into a single logarithm or vice versa.

### Which Logarithm Formula is Used to Change the Base of a Logarithm?

The change of base formula (which is one of the log formulas) is used to change the base. Using this formula, log_{b} a = (log_{c} a) / (log_{c} b). We can see that the base on the left side is "b" and the bases of logarithms on the right side are equal to "c".

### What is the Use of the Change of Base Formula (One of the Logarithmic Formulas)?

Here is an important use of the change of base formula. Usually, the calculators have options to calculate the logarithms of numbers with base 10 and with base "e". To find the logarithms of numbers with other bases than 10 and "e", we use the change of base formula. For example log_{2} 3 = (log 3) / (log 2).

### What are Natural Logarithmic Formulas?

The natural logarithm formulas are as same as that of logarithmic formulas except "log with some base" is replaced with "ln". Here are the most important natural log formulas.

- ln (xy) = ln x + ln y
- ln (x / y) = ln x - ln y
- ln a
^{x}= x ln a

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