How do I find log 10 526

3.3 logarithms

From online math bridging course 1

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Content:

  • Logarithms
  • The laws of logarithms

Learning goals:

After this section you should be able to:

  • Calculate with bases and exponents
  • Know the meaning of the expressions \ displaystyle \ ln, \ displaystyle \ lg, \ displaystyle \ log and \ displaystyle \ log_ {a}.
  • Calculate simple logarithms with the definition of the logarithm.
  • Know that logarithms are only defined for positive numbers.
  • Know the meaning of the number \ displaystyle e.
  • Use the laws of logarithms to simplify logarithmic expressions.
  • Know when the logarithmic laws are valid.
  • Change the base of logarithms

Are you still very familiar with the learning objectives from school and do you know exactly how to do the calculations? Then you can do the same with the exams begin (you can find the link in the Student Lounge).

A - base 10 logarithm

Often one uses powers with the base \ displaystyle 10 to write large numbers, for example:

\ displaystyle \ begin {align *}

10 ^ 3 & = 10 \ times 10 \ times 10 = 1000 \ ,, \ 10 ^ {- 2} & = \ frac {1} {10 \ times 10} = \ frac {1} {100} = 0 \ textrm {.} 01 \, \ mbox {.} \ end {align *}

If you look at the exponent, you can see that:

"the exponent of 1000 is 3", or that
"the exponent of 0.01 is -2".

The logarithm is defined in the same way. We wrote more formally:

"The logarithm of 1000 is 3 ". This is written \ displaystyle \ lg 1000 = 3,
"The logarithm from 0.01 is -2 ". This is written \ displaystyle \ lg 0 \ textrm {.} 01 = -2.

The following applies more generally:

The logarithm of a number \ displaystyle y is called \ displaystyle \ lg y and is the exponent that makes up the equation
\ displaystyle 10 ^ {\ \ bbox [# AAEEFF, 2pt] {\, \ phantom {a} \,}} = y \, \ mbox {.}

Fulfills. \ displaystyle y must be a positive number so that the logarithm \ displaystyle \ lg y should be defined after a power with a positive base (like 10) is always positive.

example 1

  1. \ displaystyle \ lg 100000 = 5 \ quad because \ displaystyle 10 ^ {\, \ bbox [# AAEEFF, 1pt] {\ scriptstyle \, 5 \ vphantom {,} \,}} = 100 \, 000
  2. \ displaystyle \ lg 0 \ textrm {.} 0001 = -4 \ quad because \ displaystyle 10 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, - 4 \ vphantom {,} \,}} = 0 \ textrm {.} 0001
  3. \ displaystyle \ lg \ sqrt {10} = \ frac {1} {2} \ quad because \ displaystyle 10 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 1/2 \,}} = \ sqrt {10}
  4. \ displaystyle \ lg 1 = 0 \ quad because \ displaystyle 10 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 0 \ vphantom {,} \,}} = 1
  5. \ displaystyle \ lg 10 ^ {78} = 78 \ quad because \ displaystyle 10 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 78 \ vphantom {,} \,}} = 10 ^ {78}
  6. \ displaystyle \ lg 50 \ approx 1 \ textrm {.} 699 \ quad because \ displaystyle 10 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 1 \ textrm {.} 699 \,}} \ approx 50
  7. \ displaystyle \ lg (-10) does not exist because \ displaystyle 10 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, a \ vphantom {b,} \,}} can never become -10, no matter how to choose \ displaystyle a

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Example 2

  1. \ displaystyle 10 ^ {\ textstyle \, \ lg 100} = 100
  2. \ displaystyle 10 ^ {\ textstyle \, \ lg a} = a
  3. \ displaystyle 10 ^ {\ textstyle \, \ lg 50} = 50

B - Different bases

You can also define logarithms for bases other than 10 (except for base 1). In this case, however, one must clearly show which number is the basis. For example, if you use base 2, you write \ displaystyle \ log _ {\, 2} and this means "the logarithm of base 2".

Example 3

  1. \ displaystyle \ log _ {\, 2} 8 = 3 \ quad because \ displaystyle 2 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 3 \ vphantom {,} \,}} = 8.
  2. \ displaystyle \ log _ {\, 2} 2 = 1 \ quad because \ displaystyle 2 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 1 \ vphantom {,} \,}} = 2.
  3. \ displaystyle \ log _ {\, 2} 1024 = 10 \ quad because \ displaystyle 2 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 10 \ vphantom {,} \,}} = 1024.
  4. \ displaystyle \ log _ {\, 2} \ frac {1} {4} = -2 \ quad because \ displaystyle 2 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, - 2 \ vphantom {,} \,}} = \ frac {1} {2 ^ 2} = \ frac {1} {4}.

The calculations with bases other than 2 are quite similar.

Example 4

  1. \ displaystyle \ log _ {\, 3} 9 = 2 \ quad because \ displaystyle 3 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 2 \ vphantom {,} \,}} = 9.
  2. \ displaystyle \ log _ {\, 5} 125 = 3 \ quad because \ displaystyle 5 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 3 \ vphantom {,} \,}} = 125.
  3. \ displaystyle \ log _ {\, 4} \ frac {1} {16} = -2 \ quad because \ displaystyle 4 ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, - 2 \ vphantom {,} \,}} = \ frac {1} {4 ^ 2} = \ frac {1} {16}.
  4. \ displaystyle \ log _ {\, b} \ frac {1} {\ sqrt {b}} = - \ frac {1} {2} \ quad because \ displaystyle b ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt ] {\, - 1/2 \,}} = \ frac {1} {b ^ {1/2}} = \ frac {1} {\ sqrt {b}} (if \ displaystyle b> 0 and \ displaystyle b \ not = 1).

When calculating to base 10, one rarely writes \ displaystyle \ log _ {\, 10}, but one simply writes lg or log.

C - The natural logarithm

The two most commonly used logarithms are those with a base of 10, and the number \ displaystyle e \ displaystyle ({} \ approx 2 \ textrm {.} 71828 \ ldots \,). The logarithms to the base e become natural logarithms called. Instead of \ displaystyle \ log _ {\, e}, write \ displaystyle \ ln when calculating natural logarithms.

Example 5

  1. \ displaystyle \ ln 10 \ approx 2 {,} 3 \ quad because \ displaystyle e ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 2 {,} 3 \,}} \ approx 10.
  2. \ displaystyle \ ln e = 1 \ quad because \ displaystyle e ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 1 \ vphantom {,} \,}} = e.
  3. \ displaystyle \ ln \ frac {1} {e ^ 3} = -3 \ quad because \ displaystyle e ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, - 3 \ vphantom {,} \,} } = \ frac {1} {e ^ 3}.
  4. \ displaystyle \ ln 1 = 0 \ quad because \ displaystyle e ^ {\ scriptstyle \, \ bbox [# AAEEFF, 1pt] {\, 0 \ vphantom {,} \,}} = 1.
  5. If \ displaystyle y = e ^ {\, a} then \ displaystyle a = \ ln y.
  6. \ displaystyle e ^ {\, \ bbox [# AAEEFF, 1pt] {\, \ ln 5 \ vphantom {,} \,}} = 5
  7. \ displaystyle e ^ {\, \ bbox [# AAEEFF, 1pt] {\, \ ln x \ vphantom {,} \,}} = x

Most good calculators can calculate 10 logarithms and natural logarithms.

D - laws of logarithms

In 1617-1624 Henry Biggs published a table with all the logarithms of the numbers up to 20,000, and in 1628 Adriaan Vlacq expanded the table with numbers up to 100,000 Can add logarithms of the two numbers and then calculate the number from the logarithm (this is much more effective than multiplying the numbers directly).

Example 6

Calculate \ displaystyle \ .35 \ cdot 54.

If we know that \ displaystyle 35 \ approx 10 ^ {\, 1 \ textrm {.} 5441} and \ displaystyle 54 \ approx 10 ^ {\, 1 \ textrm {.} 7324} (i.e. \ displaystyle \ lg 35 \ approx 1 \ textrm {.} 5441 and \ displaystyle \ lg 54 \ approx 1 \ textrm {.} 7324), we can easily calculate the product:

\ displaystyle

35 \ cdot 54 \ approx 10 ^ {\, 1 \ textrm {.} 5441} \ cdot 10 ^ {\, 1 \ textrm {.} 7324} = 10 ^ {\, 1 \ textrm {.} 5441 + 1 \ textrm {.} 7324} = 10 ^ {\, 3 \ textrm {.} 2765} \ ,.

If we also know that \ displaystyle 10 ^ {\, 3 \ textrm {.} 2765} \ approx 1890 (i.e. \ displaystyle \ lg 1890 \ approx 3 \ textrm {.} 2765) we have made it, the product

\ displaystyle 35 \ cdot 54 = 1890

can only be calculated with the addition of the exponents \ displaystyle 1 \ textrm {.} 5441 and \ displaystyle 1 \ textrm {.} 7324.

This is an example of the logarithmic laws, viz

\ displaystyle \ log (ab) = \ log a + \ log b

This comes from the rules of arithmetic for powers. On the one hand we have

\ displaystyle

a \ cdot b = 10 ^ {\ textstyle \ log a} \ cdot 10 ^ {\ textstyle \ log b} = 10 ^ {\, \ bbox [# AAEEFF, 1pt] {\, \ log a + \ log b \, }}

but on the other hand we also have

\ displaystyle

a \ cdot b = 10 ^ {\, \ bbox [# AAEEFF, 1pt] {\, \ log (ab) \,}} \, \ mbox {.}

With the calculation rules for powers one can similarly derive the following logarithmic laws:

\ displaystyle \ begin {align *}

\ log (ab) & = \ log a + \ log b, \ [4pt] \ log \ frac {a} {b} & = \ log a - \ log b, \ [4pt] \ log a ^ b & = b \ cdot \ log a \, \ mbox {.} \ \ end {align *}

The logarithmic laws apply regardless of the base.

Example 7

  1. \ displaystyle \ lg 4 + \ lg 7 = \ lg (4 \ cdot 7) = \ lg 28
  2. \ displaystyle \ lg 6 - \ lg 3 = \ lg \ frac {6} {3} = \ lg 2
  3. \ displaystyle 2 \ cdot \ lg 5 = \ lg 5 ^ 2 = \ lg 25
  4. \ displaystyle \ lg 200 = \ lg (2 \ cdot 100) = \ lg 2 + \ lg 100 = \ lg 2 + 2

Example 8

  1. \ displaystyle \ lg 9 + \ lg 1000 - \ lg 3 + \ lg 0 {,} 001 = \ lg 9 + 3 - \ lg 3 - 3 = \ lg 9- \ lg 3 = \ lg \ displaystyle \ frac {9 } {3} = \ lg 3
  2. \ displaystyle \ ln \ frac {1} {e} + \ ln \ sqrt {e} = \ ln \ left (\ frac {1} {e} \ cdot \ sqrt {e} \, \ right) = \ ln \ left (\ frac {1} {(\ sqrt {e} \,) ^ 2} \ cdot \ sqrt {e} \, \ right) = \ ln \ frac {1} {\ sqrt {e}}
    \ displaystyle \ phantom {\ ln \ frac {1} {e} + \ ln \ sqrt {e}} {} = \ ln e ^ {- 1/2} = - \ frac {1} {2} \ cdot \ ln e = - \ frac {1} {2} \ cdot 1 = - \ frac {1} {2} \ vphantom {\ biggl (}
  3. \ displaystyle \ log_2 36 - \ frac {1} {2} \ log_2 81 = \ log_2 (6 \ cdot 6) - \ frac {1} {2} \ log_2 (9 \ cdot 9)
    \ displaystyle \ phantom {\ log_2 36 - \ frac {1} {2} \ log_2 81} {} = \ log_2 (2 \ cdot 2 \ cdot 3 \ cdot 3) - \ frac {1} {2} \ log_2 ( 3 \ times 3 \ times 3 \ times 3)
    \ displaystyle \ phantom {\ log_2 36 - \ frac {1} {2} \ log_2 81} {} = \ log_2 (2 ^ 2 \ cdot 3 ^ 2) - \ frac {1} {2} \ log_2 (3 ^ 4) \ vphantom {\ Bigl (}
    \ displaystyle \ phantom {\ log_2 36 - \ frac {1} {2} \ log_2 81} {} = \ log_2 2 ^ 2 + \ log_2 3 ^ 2 - \ frac {1} {2} \ log_2 3 ^ 4
    \ displaystyle \ phantom {\ log_2 36 - \ frac {1} {2} \ log_2 81} {} = 2 \ log_2 2 + 2 \ log_2 3 - \ frac {1} {2} \ cdot 4 \ log_2 3
    \ displaystyle \ phantom {\ log_2 36 - \ frac {1} {2} \ log_2 81} {} = 2 \ cdot 1 + 2 \ log_2 3 - 2 \ log_2 3 = 2 \ vphantom {\ Bigl (}
  4. \ displaystyle \ lg a ^ 3 - 2 \ lg a + \ lg \ frac {1} {a} = 3 \ lg a - 2 \ lg a + \ lg a ^ {- 1}
    \ displaystyle \ phantom {\ lg a ^ 3 - 2 \ lg a + \ lg \ frac {1} {a}} {} = (3-2) \ lg a + (-1) \ lg a = \ lg a - \ lg a = 0

E - change base

Sometimes you want to write logarithms in one base than logarithms in another base.

Example 9

  1. Write \ displaystyle \ lg 5 as a natural logarithm.

    By definition, \ displaystyle \ lg 5 is the number that makes up the equation
    \ displaystyle 10 ^ {\ lg 5} = 5 \, \ mbox {.}

    Fulfills. By calculating the natural logarithm of both sides, we get

    \ displaystyle \ ln 10 ^ {\ lg 5} = \ ln 5 \, \ mbox {.}

    With the logarithmic law \ displaystyle \ ln a ^ b = b \ ln a we write the left side as \ displaystyle \ lg 5 \ cdot \ ln 10 and get the equation

    \ displaystyle \ lg 5 \ cdot \ ln 10 = \ ln 5 \, \ mbox {.}

    Division by \ displaystyle \ ln 10 gives the answer

    \ displaystyle

    \ lg 5 = \ frac {\ ln 5} {\ ln 10} \ qquad (\ approx 0 \ textrm {.} 699 \ ,, \ quad \ text {so} \ 10 ^ {0 \ textrm {.} 699 } \ approx 5) \, \ mbox {.}

  2. Write the 2-logarithm of 100 as a 10-logarithm, lg.

    According to the definition of the logarithm, it is clear that \ displaystyle \ log_2 100 is the equation
    \ displaystyle 2 ^ {\ log _ {\ scriptstyle 2} 100} = 100

    Fulfills. We log both sides (using the 10 logarithm) and get

    \ displaystyle

    \ lg 2 ^ {\ log _ {\ scriptstyle 2} 100} = \ lg 100 \, \ mbox {.}

    After \ displaystyle \ lg a ^ b = b \ lg a, we get \ displaystyle \ lg 2 ^ {\ log_2 100} = \ log _ {\ scriptstyle 2} 100 \ cdot \ lg 2 and the right side is just \ displaystyle \ lg 100 = 2. This gives the equation

    \ displaystyle

    \ log _ {\ scriptstyle 2} 100 \ cdot \ lg 2 = 2 \, \ mbox {.}

    Division by \ displaystyle \ lg 2 shows that

    \ displaystyle

    \ log _ {\ scriptstyle 2} 100 = \ frac {2} {\ lg 2} \ qquad ({} \ approx 6 \ textrm {.} 64 \ ,, \ quad \ text {so is} \ 2 ^ {6 \ textrm {.} 64} \ approx 100) \, \ mbox {.}

The general formula for changing the base from \ displaystyle a to \ displaystyle b in logarithms is

\ displaystyle

\ log _ {\ scriptstyle \, a} x = \ frac {\ log _ {\ scriptstyle \, b} x} {\ log _ {\ scriptstyle \, b} a} \, \ mbox {.}

For example, if we want to write \ displaystyle 2 ^ 5 to base 10, we first write 2 to base 10

\ displaystyle 2 = 10 ^ {\ lg 2}

and use the calculation rules for powers

\ displaystyle

2 ^ 5 = (10 ^ {\ lg 2}) ^ 5 = 10 ^ {5 \ cdot \ lg 2} \ quad ({} \ approx 10 ^ {1 \ textrm {.} 505} \,) \, \ mbox {.}

Example 10

  1. Write \ displaystyle 10 ^ x to the natural base e.

    First we write 10 to the base e,
    \ displaystyle 10 = e ^ {\ ln 10}

    and use the calculation rules for powers

    \ displaystyle

    10 ^ x = (e ^ {\ ln 10}) ^ x = e ^ {\, x \ cdot \ ln 10} \ approx e ^ {2 \ textrm {.} 3 x} \, \ mbox {.}

  2. Write \ displaystyle e ^ {\, a} to base 10

    The number \ displaystyle e can be written like \ displaystyle e = 10 ^ {\ lg e}, and therefore is
    \ displaystyle

    e ^ a = (10 ^ {\ lg e}) ^ a = 10 ^ {\, a \ cdot \ lg e} \ approx 10 ^ {\, 0 \ textrm {.} 434a} \, \ mbox {.}

Any questions about this chapter? Then take a look in the course forum (you can find the link in the Student Lounge) or ask ombTutor via Skype

No more questions? Then go on with that Exercises .