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LOGARITHMS

Definition

Common logarithms

The three laws of logarithms

Change of base

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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Do the problem yourself first!

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.

Problem 2. Write each of the following in logarithmic form.

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a) bⁿ = x	log_bx = n	b) 2³ = 8	log₂8 = 3

c) 10² = 100	log₁₀100 = 2	d) 5⁻² = 1/25.	log₅1/25 = −2.

Problem 3. Write each of the following in exponential form.

a) log_bx = n	bⁿ = x	b) log₂32 = 5	2⁵ = 32

c) 2 = log₈64	8² = 64	d) log₆1/36 = −2	6⁻² = 1/36

Problem 4. Evaluate the following.

a) log₂16	= 4	b) log₄16	= 2

c) log₅125	= 3	d) log₈1	= 0

e) log₈8	= 1	f) log₁₀1	= 0

Problem 5. What number is n?

a) log₁₀n = 3	1000	b) 5 = log₂n	32

c) log₂n = 0	1	d) 1 = log₁₀n	10

e) log_n	1 16	= −2	4	f) log_n	1 5	= −1	5

g) log₂	1 32	= n	−5	h) log₂	1 2	= n	−1

Problem 6. log_bb^x = x

Problem 7. Evaluate the following.

a) log₉

1
9

b) log₉	1 81	= −2	c) log₂	1 4	= −2

d) log₂	1 8	= −3	e) log₂	1 16	= −4

f) log₁₀ .01	= −2		g) log₁₀ .001	= −3

Common logarithms

The system of common logarithms has 10 as its base. When the base is not indicated,

log 100 = 2

then the system of common logarithms -- base 10 -- is implied.

Here are the powers of 10 and their logarithms:

Powers of 10:	1 1000	1 100	1 10	1	10	100	1000	10,000

Logarithms:	−3	−2	−1	0	1	2	3	4

Logarithms replace a geometric series with an arithmetic series.

Problem 8. log 10ⁿ = ? n. The base is 10.

Problem 9. log 58 = 1.7634. Therefore, 10^1.7634 = ?

Problem 10. log (log x) = 1. What number is x?

The three laws of logarithms

1. log_bxy = log_bx + log_by

"The logarithm of a product is equal to the sum
of the logarithms of each factor."

2. log_b

= log_bx − log_by

"The logarithm of a quotient is equal to the logarithm of the numerator
minus the logarithm of the denominator."

3. log_b xⁿ = n log_bx

"The logarithm of x with a rational exponent is equal to
the exponent times the logarithm."

We will prove these laws below.

Example 8. Apply the laws of logarithms to log

abc²
d³

Answer. According to the first two laws,

log	abc² d ³	=	log (abc²) − log d ³

		=	log a + log b + log c² − log d ³

		=	log a + log b + 2 log c − 3 log d,

according to the third law.

The Answer above shows the complete theoretical steps. In practice, however, it is not necessary to write the line

log

abc²
d ³

log (abc²) − log d ³

The student should be able to go immediately to the next line --

log

abc²
d ³

log a + log b + log c² − log d ³

-- if not to the very last line

log abc²
d ³ = log a + log b + 2 log c − 3 log d.

Example 9. Solve this equation for x:

log 3^{2x + 5}	=	1

Solution. According to the 3rd Law, we may write

(2x + 5)log 3	=	1

Now, log 3 is simply a number. Therefore, on distributing log 3,

2x· log 3 + 5 log 3	=	1

2x· log 3	=	1 − 5 log 3

x	=	1 − 5 log 3 2 log 3

By this technique, we can solve equations in which the unknown appears in the exponent.

Problem 11. Use the laws of logarithms to rewrite the following.

a) log	ab c	= log a + log b − log c

b) log	ab² c⁴	= log a + 2 log b − 4 log c

Change of base

Say that we know the values of logarithms of base 10, but not, for example, in base 2. Then we can convert a logarithm in base 10 to one in base 2 -- or any other base -- by realizing that the values will be proportional.

Each value in base 2 will differ from the value in base 10 by the same constant k.

Now, to find that constant, we know that

Therefore, on putting x = 2 above:

That implies

Therefore,

Logarithms

That is,

By knowing the values of logarithms in base 10, we can in this way calculate their values in base 2.

In general, then, if we know the values in base a, then the constant of proportionality in changing to base b, is the reciprocal of its log in base a.

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