**Definition**

Let , *a*, *b* > 0 and *b* ≠ 1. The logarithm of *a* to base *b*, denoted by , is defined as follows.

if and only if

The number* b* is called the * base* of the logarithm, and

*a*is called the

*.*

**argument**

**Notes**

Throughout this post, if the base of the logarithm is not stated, it is assumed that the base is 10. Thus, . Logarithms to base 10 are called * common logarithms* or

*. Another type of logarithm that is often used is the logarithm to base*

**Briggsian logarithms***e*, where

*e*is the Euler number. It is called the

*or the*

**natural logarithm***. The logarithm has a special notation, namely*

**Napierian logarithm****ln**. For every

*a*> 0, ln

*a*= log

_{e}

*a*.

**Examples**

log_{2} 16 = 4 (because 2^{4} = 16)

log_{5} 125 = 3 (because 5^{3} = 125)

log_{5} .2 = -1 (because 5^{-1} = .2)

log 10000 = 4 (because 10^{4} = 10000)

ln 1 = 0 (because *e*^{0} = 1)

By the definition above, the base of a logarithm cannot be 1. Why? If there was not such restriction, some indeterminacy would exist. To demonstrate this, log_{1} 1 = 4 as 1^{4} = 1. On the other hand, it also holds that log_{1} 1 = 2 (since 1^{2} = 1). Then, it follows that 4 = 2, contradicting 4 ≠ 2. Therefore, the base of a logarithm must not be equal to 1.

Going back to the definition of logarithms, take 2^{3} = 8. By the definition, it follows that 3 = log_{2} 8. By substituting log_{2} 8 for 3 in 2^{3} = 8, it follows that . This is an example of how Property 1 applies.

**Property 1**

; *a*, *b* > 0 and *b* ≠ 1

**Example 1**

Compute .

**Answer**

**Example 2**

Compute .

**Answer**

**Property 2**

; *b*, *x*, *y* > 0 and *b* ≠ 1

**Example 3**

Suppose that log 2 = 0.3010 and log 3 = 0.4771. Compute log 6.

**Answer**

log 6 = log 2⋅3 = log 2 + log 3 = 0.3010 + 0.4771 = 0.7781

**Example 4**

Let log 2 = 0.3010. Compute log 2000.

**Answer**

log 2000 = log 2⋅1000 = log 2 + log 1000 = 0.3010 + 3 = 3.3010

**Property 3**

; *b*, *x*, *y* > 0 and *b* ≠ 1

**Example 5**

Suppose that log 2 = 0.3010 and log 3 = 0.4771. Compute log 1.5.

**Answer**

**Example 6**

Let log 2 = 0.3010. Compute log 5.

**Answer**

**Property 4**

; *a*, *b* > 0 and *b* ≠ 1 and

**Example 7**

Let log 3 = 0.4771. Compute log 243.

**Answer**

log 243 = log 3^{5} = 5⋅log 3 = 5⋅0.4771 = 2.3855

**Example 8**

Compute ln *e*^{√7}.

**Answer**

ln *e*^{√7} = (√7) ⋅ ln *e* = √7 ⋅ 1 = √7 [Recall that ln *e* = log_{e} *e* = 1.]

**Example 9**

Compute .

**Answer**

**Property 5**

; , *n* ≠ 0, *a*, *b* > 0 and *b* ≠ 1

**Example 10**

Compute log_{16} 64.

**Answer**

**Example 11**

Let log_{2} 3 = 1.585. Compute log_{8} 3.

**Answer**

**Property 6**

; *a*, *b* > 0 and *b* ≠ 1

In this formula, *k* is any positive real number and *k* ≠ 1.

**Example 12**

Compute log_{16} 64.

**Answer**

(cf. Example 10 above.)

**Example 13**

Let log_{2} 3 = 1.585. Compute log_{3} 2.

**Answer**

**Property 7**

; *a*, *b* > 0 and *a*, *b* ≠ 1.

It can be easily seen that the formula in Property 7 also addresses the problem in Example 13 .