Logarithmic Equations: Formulas, Properties, Examples of Problems and Their Discussion

Logarithmic Equations: Formulas, Properties, Examples of Problems and their Discussion – What is a Logarithmic Equation and an example of a problem? On this occasion, Seputarknowledge.co.id will discuss it and of course about other things that also cover it. Let's look at the discussion together in the article below to better understand it.

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Logarithmic Equations: Formulas, Properties, Examples of Problems and Their Discussion

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A logarithm is a mathematical operation that is the reciprocal (or inverse) of an exponential or exponential power. In this formula, a is the base or principal of the logarithm. Judging from the origin of the words, the word Algorithm has a rather strange history. People only find the word Algorism which means the process of calculating with Arabic numerals.

Logarithmic equationa is an equation whose variable is a numerus or logarithmic base number. Logarithms can also be interpreted as mathematical operations which are the opposite (or inverse) of exponents or exponents.

Someone is said to be "Algorist" when calculating using Arabic numerals. Linguists have tried to find the origin of this word, but the results have been unsatisfactory. Finally, the historians of mathematics found the origin of the word, which comes from the name of the author of the book Famous Arabic, namely Abu Abdullah Muhammad Ibn Musa Al-Khuwarrismi read by westerners to be Algorism.

The inventor was a mathematician from Uzbeskitan named Abu Abdullah Muhammad Ibn Musa Al-Khwarizmi. In western literature, he is better known as Algorism. This call is then used to refer to the algorithm concept he found.

Abu Abdullah Muhammad Ibn Musa Al-Khuwarizmi (770-840) was born in Khawarizm (Kheva), a city south of the Oxus river (now Uzbekistan) in 770 AD. His parents then moved to a place south of Baghdad (Iraq), when he was still small.

A work using Indian numerals, which was translated and used for the first time in the west, is entitled al-jam' wa'l-tafriq bi hisab al-hind (Addition and Substraction in Indian Arithmetics). The book is the glorious work of the Muslim mathematician Muhammad ibn Musa Al-Khwarismi (780-850M).

John Napier was an English mathematician, born at Merchiston Castle Eidenburg. Napier finished school in France at the age of 13, then he went on to the University of St. Andrews in Scotland.

In 1612 AD, he discovered a system that was named "logarithm" which was derived from the name khawarizmi. Now his findings are better known as Napier logarithms (Napierian Logarithms).

Napier once made tables carved into bone-like ivory. Then, they named it after Napier's Bones (Napier's Bones).

When Napier's book on logarithms was published in 1614, it amazed scientists as much as the modern calculator invented.

With the help of logarithms they can do difficult multiplication and division in a fast and easy way for the first time. Napier spent his life tinkering with mathematics.

He died in 1617 at the age of 67 and was buried in Edinburgh. (Johanes, et al: 33).

Because it was not pleasant to see the base numbers used in logarithms at that time, Henry Briggs (British mathematician) made a general table of logarithms (The Table of Common Logarithms) with base 10 numbers immediately after that.

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Logarithmic Formulas

a^c = b → ª log b = c

Information:

a = base
b = dilogarithmic number
c = logarithmic result

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Properties of Logarithms

ª log a = 1

ª log 1 = 0

ª log aⁿ = n

ª log bⁿ = n • ª log b

ª log b • c = ª log b + ª log c

ª logs b/c = ª log b – ª log c

ªˆⁿ log b m = m/n • ª log b

ª log b = 1 ÷ b log a

ª log b • b log c • c log d = ª log d

ª log b = c log b ÷ c log a

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Properties - Properties of Logarithmic Equations

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  Logarithmic Properties of Multiplication:

A logarithm is the sum of two other logarithms whose second numerus is a factor of the initial numerus.

^alog p. q = ^alog p+ ^alog q

With the condition that is = a > 0, a \ne 1, p > 0, q > 0.

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  Logarithmic Multiplication :

A logarithm a can be multiplied by logarithm b if the numerus value of logarithm a is the same as the base number of logarithm b. The result of the multiplication is the new logarithm with the base number value equal to the logarithm a, and the numerus value equal to the logarithm b.

^alog b x ^blogc = ^alog c

With the condition that is = a > 0, a \ne 1.

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  Logarithmic Properties Of Division:

A logarithm is the result of subtracting two other logarithms whose second numerus is a fraction or division of the initial logarithm's numerus value.

With the conditions being = a > 0, a\ne 1, p > 0, q > 0.

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  Properties of Inverse Logarithms:

A logarithm is inversely proportional to another logarithm that has the values ​​of the base and numerus interchanging.

^alogb = 1/^blog a

Provided that = a > 0, a \ne 1.

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  Logarithm of Opposite Sign :

A logarithm of opposite sign to a logarithm has a numerus, which is an inverted fraction of the initial logarithm's numerus value.

^alog p/q = – ^alog p/q

With the conditions being = a > 0, a\ne 1, p > 0, q > 0.

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  Logarithmic Properties of Exponentials:

A logarithm, that is, with its numerus value, is an exponent (power) and can be used as a new logarithm by removing the exponent as a multiplier.

^alog b^p.s = p. ^alog b

Provided that = a > 0, a \ne 1, b > 0

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  Logarithmic Base Numbers:

A logarithm that is with the value of the base number is an exponent (power) which can be used as a new logarithm by removing the exponent as the divisor.

a^p.slogb = 1/p^alog b

Provided that = a > 0, a \ne 1.

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  Logarithmic Principal Numbers Comparable to Numeral Powers:

A logarithm that is the value of its numerus is an exponent (power) of the value of the base number which has the same result as the value of the power of the numerus.

^alog a^p.s = p

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  Exponential Logarithm :

A number that has a logarithmic exponent, the result of the exponent is the numerical value of the logarithm.

a ^alog m = m

With conditions that are = a > 0, a \ne 1, m > 0.

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  Changing the Logarithmic Base :

A logarithm can also be broken down into the ratio of two logarithms.

^p.slog q = ^alog p/^a log q

With the conditions being = a > 0, a\ne 1, p > 0, q > 0

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Logarithm example

Logarithms also have their own examples of numbers, which are as follows:

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Examples of Logarithmic Equation Problems

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Problem 1

Know the logarithm ³log 5 = x and ³log 7 = y. then, the value of ³log 245 1/2 is….

Resolution:

Problem 2

1. Value of ²logs 4+ ²logs 12 – ²log 6 =…

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1. 8
2. 6
3. 5
4. 4
5. 3

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Discussion :

For questions like the one above, we need to remember the nature of logarithms

^alog(b.c) = ^alog b+ ^alog c, And

^alogs  = ^alog b – ^alog c

so, to solve the problem above, we use the two properties of logarithms. Where the calculation will be:

²logs 4+ ²logs 12 – ²log 6 = ²logs

= ²logs 8

Then, for the final solution, we need to remember the following properties, namely:

^alogs  = n. ^alog b

→ 8 =

So, the final solution will be like this:

²log 8 = ²logs

= 3. ²log 2 → don't forget this one: ^aloga = 1

= 3. 1

= 3 ( E )

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Problem 3

If log 3 = 0.4771 and log 2 = 0.3010, then the value of log 75 =...

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1. 0,7781
2. 0,9209
3. 1,0791
4. 1,2552
5. 1,8751

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Discussion :

For problems with a model like this, there is a key to the process that we must understand. Namely is a description that shows the value of log 2 and log 3. With this additional information, meaning that should be on our minds is how to convert log 75 into logarithmic form containing elements of numbers 2 and 3.

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→ 75 = 3. 25 = 3 .

So, if we change the number 75 with 3., then we will get:

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logs 75 = logs ( 3. ) → with this, we have to remember the properties: ^alog(b.c) = ^alog b+ ^alog c

= log 3 + log → don't forget that: ^alogs  = n. ^alog b

= log 3 + 2. logs 5

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The point is to change the number 5 in log 5, because in the question the explanations are log 2 and log 3, while log 5 is not given any information.

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For that, the tricks that need to be done here are:

→ 5 =

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We need to change the number 5 into a number that contains elements of number 2 and its value does not change (still has a value of 5). So, if we solve, it will be:

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log 75 = log 3 + 2. log → certainly still remember the properties ^alogs  = ^alog b – ^alog c, right?

= log 3 + 2 ( log 10 – log 2 ) → log 10 = ¹⁰log 10 = 1 → ^aloga = 1

= 0,4771 + 2 ( 1 – 0,3010 )

= 1.8751 ( E )

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Problem 4

Is known ²log 3 = 1.6 and ²log 5 = 2.3; value from ²logs ..

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1. 10,1
2. 6,9
3. 5,4
4. 3,2
5. 3,7

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Discussion :

Slightly similar to the previous problem, by knowing there is a description in the matter of the value of a logarithm of a number, then what we need to do is to change it into a form that contains the number elements that match the description.

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→ 125 = 5. 5. 5 =

→ 9 =

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So, if we solve the problem, it will be:

²logs = ²log → predictable right? Here we need properties: ^alogs  = ^alog b – ^alog c

= ²logs – ²logs

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Then, the logarithmic property that we use next is the property:

^alogs  = n. ^alog b

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then, the above equation will then become:

= 3. ²logs 5 – 2. ²logs 3

= 3. ( 2,3 ) – 2. ( 1,6 )

= 6,9 – 3,2

= 3.7 ( E )

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