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Question

i need help with logarithems how do you do them?

Answer 1

20

LOGARITHMS

Definition

Common logarithms

Natural logarithms

The three laws of logarithms

Proof of the laws of logarithms



A LOGARITHM is an exponent.

Since

23 = 8,

then 3 is called the logarithm of 8 with base 2. We write

3 = log28.

3 is the exponent to which 2 must be raised to produce 8.

We write the base 2 as a subscript.

Thus a logarithm is the exponent to which the base must be raised to produce a given number.

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:



logbx = 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.

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.

logb1 = 0



Example 4. Evaluate log55.

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

log55 = 1.

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

logbb = 1



Example 5. log22m = ?

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

log22m = m.

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

logbbx = 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 bx.

The rule also shows that the inverse of the function logbx is the exponential function 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.

Which numbers, then, will have negative logarithms?

Proper fractions

Example 7. log2 .25 = ?

Answer. .25 = ¼ = 2−2. Therefore,

log2 .25 = log22−2 = −2.

Example 8. log3 = ?

Answer. = 31/5. (Definition of a rational exponent.) Therefore,

log3 = log331/5 = 1/5.

Problem 1. Write each of the following in logarithmic form.

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

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

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

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

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

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

Problem 3. Evaluate the following.

a) log216 = 4 b) log416 = 2

c) log5125 = 3 d) log81 = 0

e) log88 = 1 f) log101 = 0

Problem 4. What number is n?

a) log10n = 3 1000 b) 5 = log2n 32

c) log2n = 0 1 d) 1 = log10n 10
e) logn 1
16 = −2 4 f) logn 1
5 = −1 5

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

Problem 5. logbbx = x

Problem 6. Evaluate the following.

a) log9 1
9 = log99−1 = −1
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) logb = 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, 101.7634 = ?

58. 1.7634 is the common logarithm of 58. When 10 is raised to that exponent, 58 is produced.

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

log a = 1, implies a = 10. (See above.) Therefore, log (log x) = 1 implies log x = 10. Since 10 is the base,

x = 1010 = 10,000,000,000



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.

ex has the simplest derivative. See Lesson 14 of An Approach to Calculus.)

e can be calculated from the following series involving factorials:

e = 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 use the notation "ln."

ln x means logex.

Problem 10. What number is ln e ?

ln e = 1. The logarithm of the base itself is always 1. e is the base.



Problem 11. Write in exponential form (Example 1): y = ln x.

ey = x.

e is the base.



The three laws of logarithms

1. logbxy = logbx + logby

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

2. logb x
y = logbx − logby

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



3. logb x n = n logbx

"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 1. Use the laws of logarithms to rewrite log
z5 .

Answer. According to the first two laws,

log
z5 = log x + log − log z5

Now, = y½. Therefore, according to the third law,

log
z5 = log x + ½ log y − 5 log z

Example 2. 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 3. Solve this equation for x:

log 32x + 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 multiplying out by 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· x1/3 ln x)

= ½ (ln cos x + 1/3 ln x + ln ln x)
f) ln (a2x − 1 b5x + 1 ) = ln a2x − 1 + ln b5x + 1

= (2x − 1) ln a + (5x + 1) ln b

Problem 13. Solve for x.

ln 23x + 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 = 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 = ep + q.

And therefore

ln ab = ln ep + 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.


I hope this helps you

Answer 2

I suggest you post a problem you are wrestling with here to get some help

Answer 3

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I understand u r bored out of your mind but R u writing a book about LOGS ?

LMAO

Answer 4

What do u need in logarithm???Give some extra details...

Answer 5

I would tell you, but there's no space left