LOGARITHM

WHEN WE ARE GIVEN the base 2, for example, and exponent 3, then we can evaluate 2³.

2³ = 8.

Inversely, if we are given the base 2 and its power 8 --

2^? = 8

-- then what is the exponent that will produce 8?

That exponent is called a logarithm.  We call the exponent 3 the logarithm of 8 with base 2.  We write

3 = log₂8.

We write the base 2 as a subscript.
3 is the exponent to which 2 must be raised to produce 8.
A logarithm is an exponent.

Since

10⁴ = 10,000

then

log₁₀10,000 = 4.

"The logarithm of 10,000 with base 10 is 4."

4 is the exponent to which 10 must be raised to produce 10,000.

"10⁴ = 10,000" is called the exponential form.

"log₁₀10,000 = 4" is called the logarithmic form.

Here is the definition:

log_bx = n   means   bⁿ = x.

That  base  with that  exponent  produces x.

Example 1.   Write in exponential form:   log₂32 = 5.

 Answer.   2⁵ = 32.

Example 2.   Write in logarithmic form:  4−2  =

1 16

.

Answer.   log4

1 16

= −2.

Problem 1.   Which numbers have negative logarithms?

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Proper fractions.

Example 3.   Evaluate  log₈1.

 Answer.   8 to what exponent produces 1?  8⁰ = 1.

log₈1 = 0.

We can observe that, in any base, the logarithm of 1 is 0.

log_b1 = 0

Example 4.    Evaluate  log₅5.

 Answer.   5 with what exponent will produce 5?   5¹ = 5.  Therefore,

log₅5 = 1.

In any base, the logarithm of the base itself is 1.

log_bb = 1

Example 5 .   log₂2^m = ?

 Answer.   2 raised to what exponent will produce 2^m ?   m, obviously.

log₂2^m = m.

The following is an important formal rule, valid for any base b:

log_bb^x = x

This rule embodies the very meaning of a logarithm.  x -- on the right -- is the exponent to which the base b must be raised  to produce b^x.

The rule also shows that the exponential function b^x is the inverse of the function log_bx.  We will see this in the following Topic.

Example 6 .   Evaluate  log3

1 9

.

Answer.

1 9

is equal to 3 with what exponent?

1 9

= 3−2.

log3

1 9

=

log33−2  =  −2.

Compare the previous rule.

Example 7.   log₂ .25 = ?

 Answer.   .25 = ¼ = 2⁻².  Therefore,

log₂ .25 = log₂2⁻² = −2.