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How Can The One To One Property Of Logarithms Be Used To Solve Logarithmic Equations?

How can the one-to-one property of logarithms be used to solve logarithmic equations?

Answer

A logarithm is a quantity representing the power to which a fixed number ( the base ) must be raised to produce a given number. Logarithms are a convenient way to express large numbers. The base 10 logarithm of a number is roughly the number of digits in that number.

Let’s take an example, say 2

2 ∗ 2 = 2^{2 }= 4

2 ∗ 2 ∗ 2 = 2^{3 }= 8

:

2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 = 28 = 256

It can be noticed that every time the advancement in the exponent (the little number on the top right corner of 2), the product grows much bigger. For example, advancing from 2 to 3 grows the product by 4 and from 3 to 8 by 248.

It would not be easier to keep track of the exponent rather than the product. Instead of writing 256, we can record it as exponent 8 (for the base 2), and so on.

The same rules can be used for the numbers 3, 4 and so on and we end up with a table of values relating an exponent with the product for a given base.

Why do we want to do that, because it seems cumbersome at first glance? Let us go back to the first example:

2 ∗ 2 = 2^{1 }∗ 2^{1 }= 2^{1+1 }= 4

Notice it turned a multiplication (2x2) into an addition (1+1).

This is the fundamental advantage of logarithms, because it turns an exponential operation (multiplication) into a linear one (addition), by making use of the exponent!

It is useful is such a way that it can multiply two very big numbers by adding two relatively small numbers before converting the sum into the desired product. This was one of the earliest uses of logarithms, to calculate huge numbers in astronomy.

Logarithms were quickly adopted by scientists because of various useful properties that simplified long, tedious calculations. In particular, scientists could find the product of two numbers *m* and *n* by looking up each number’s logarithm in a special table, adding the logarithms together, and then consulting the table again to find the number with that calculated logarithm known as its antilogarithm. Expressed in terms of common logarithms, this relationship is given by log *mn* = log *m* + log *n*. For example, 100 × 1,000 can be calculated by looking up the logarithms of 100 (2) and 1,000 (3), adding the logarithms together (5), and then finding its antilogarithm (100,000) in the table. Similarly, division problems are converted into subtraction problems with logarithms: log *m*/*n* = log *m* − log *n*. This is not all; the calculation of powers and roots can be simplified with the use of logarithms. Logarithms can also be converted between any positive bases, except that 1 cannot be used as the base since all of its powers are equal to 1

Only logarithms for numbers between 0 and 10 were typically included in logarithm tables. To obtain the logarithm of some number outside of this range, the number was first written in scientific notation as the product of its significant digits and its exponential power—for example, 358 would be written as 3.58 × 10^{2}, and 0.0046 would be written as 4.6 × 10^{−3}. Then the logarithm of the significant digits—a decimal fraction between 0 and 1, known as the mantissa—would be found in a table. For example, to find the logarithm of 358, one would look up log 3.58 ≅ 0.55388. Therefore, log 358 = log 3.58 + log 100 = 0.55388 + 2 = 2.55388. In the example of a number with a negative exponent, such as 0.0046, one would look up log 4.6 ≅ 0.66276. Therefore, log 0.0046 = log 4.6 + log 0.001 = 0.66276 − 3 = −2.33724.

The one - to - one property states that a function is one - to - one is and only if f ( *x *) = f ( *y *) implies that *x = y. *To solve the logarithmic equation the one to one property of is used. The one - to - one states that, for any real number *x *> 0, S > 0, T > 0 and any positive real number a, where a ≠ 1. The reason it works for logarithms is that if I assume two logs with the same base are the same, we have log ( *x* ) = log ( *y* ). Thus exponentiating both sides, the equation forms as e^{log }( *x* ) = e^{log }( *y *), and because e^{log(a) }= a in the real numbers, hence it can be conclude that *x* = *y.*

log_{a }S = log_{a }T if and only if S=T

If log_{2 }( *x *− 1 ) = log_{2 }( 8 ), then *x *– 1 = 8

So, if x−1=8

Then the solution can be solved for *x = *9. To check *x *= 9 can be substituted into the original equation:

Log_{2 }( 9 – 1 ) = Log_{2 } ( 8 ) = 3

In simple words when the logarithmic equation have the same base at the both sides, the argument must be equal. When the arguments are the algebraic expressions then also this can be applied. Therefore, when given an equation with logs of the same base on each side, now the rules of logarithms can again be written by each side as a single logarithm. Then the fact can be utilized as logarithmic function are one - to - one to set arguments equal to another and solve the unknown.

For example, consider the equation

Log ( 4*x *– 2 ) – log ( 4 ) = log ( *x *+ 4 )

To solve this equation, the rules of logarithms to used and again written as the left side as a single logarithm, and then apply the one-to-one property to solve for *x*:

To check the result, substitute *x *= 5 into