Topics in

P R E C A L C U L U S

LOGARITHMS

Definition

Common logarithms

Natural logarithms

The three laws of logarithms

Proof of the 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 =

Problem 1. Which numbers have negative logarithms?

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Do the problem yourself first!

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 inverse of the function log_bx is the exponential function b^x. 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.

Example 8. log₃ = ?

Answer. = 3^1/5. (Definition of a rational exponent.) Therefore,

log₃ = log₃3^1/5 = 1/5.

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

h) log₆	= 1/3	i) log_b	= 3/4

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 7. log 10ⁿ = ? n. The base is 10.

Problem 8. log 58 = 1.7634. Therefore, 10^1.7634 = ?

Problem 9. log (log x) = 1. What number is x?

Natural logarithms

The system of natural logarithms has the number called e as its base. (e is named after the 18th century Swiss mathematician, Leonhard Euler.) e is the base used in calculus. It is called the "natural" base because of certain technical considerations.

e^x has the simplest derivative. Lesson 14 of An Approach to Calculus.)

e can be calculated from the following series expressed with factorials:

= 1 +

1
1!

1
2!

1
3!

1
4!

+ . . .

e is an irrational number, whose decimal value is approximately

2.71828182845904.

To indicate the natural logarithm of a number we write "ln."

ln x means log_ex.

Problem 10. What number is ln e ?

Problem 11. Write in exponential form (Example 1): y = ln 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 a power of x is equal to the exponent of that power
times the logarithm of x."

We will prove these laws below.

Example 9. Apply the laws of logarithms to log

abc²
d ³

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 10.    Use the laws of logarithms to rewrite log
  z⁵ .

Answer. log
  z⁵   =  log x + log − log z⁵

Now, = y^½. (Lesson 29 of Algebra.) Therefore, according to the third law,

log

z⁵

= log x + ½ log y − 5 log z.

Example 11. Use the laws of logarithms to rewrite ln .

Solution.

ln	=	ln (sin x ln x)^½

	=	½ ln (sin x ln x), 3rd Law

	=	½ (ln sin x + ln ln x), 1st Law

Note that the factors sin x ln x are the arguments of the logarithm function.

Example 12. 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 12. 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

c) log	z	= 1/3 log x + 1/2 log y − log z

d) ln (sin²x ln x) = ln sin²x + ln ln x = 2 ln sin x + ln ln x

e) ln	= ½ ln (cos x· x^1/3 ln x)

	= ½ (ln cos x + 1/3 ln x + ln ln x)

f) ln (a^{2x − 1}b^{5x + 1})	= ln a^{2x − 1} + ln b^{5x + 1}

	= (2x − 1) ln a + (5x + 1) ln b

Problem 13. Solve for x.

ln 2^{3x + 1}	=	5

(3x + 1) ln 2	=	5

3x ln 2 + ln 2	=	5

3x ln 2	=	5 − ln 2

x	=	5 − ln 2 3 ln 2

Problem 14. Prove: −ln x

= ln

1
x

−ln x = (−1)ln x	=	ln	x⁻¹,	Third law

	=	ln	1 x

Proof of the laws of logarithms

The laws of logarithms will be valid for any base. We will prove them for base e, that is, for y = ln x.

1. ln ab = ln a + ln b.

The function y = ln x is defined for all positive real numbers x. Therefore there are real numbers p and q such that

p = ln a and q = ln b.

This implies

a = e ^p and b = e ^q.

Therefore, according to the rules of exponents,

ab = e ^p· e ^q = e^{p + q}.

And therefore

ln ab = ln e^{p + q} = p + q = ln a + ln b.

That is what we wanted to prove.

In a similar manner we can prove the 2nd law. Here is the 3rd:

3. ln aⁿ = n ln a.

There is a real number p such that

p = ln a;

that is,

a = e ^p.

And the rules of exponents are valid for all rational numbers n (Lesson 29 of Algebra; an irrational number is the limit of a sequence of rational numbers). Therefore,

aⁿ = e ^pn.

This implies

ln aⁿ = ln e ^pn = pn = np = n ln a.

That is what we wanted to prove.

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.