Logarithmic To Exponential Form

What is a logarithm?

Logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if bx = n, in which case one writes x = logb n. For example, 23 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log2 8. In the same fashion, since 102 = 100, then 2 = log10 100. Logarithms of the latter sort (that is, logarithms with base 10) are called common, or Briggsian, logarithms and are written simply log n. Invented in the 17th century to speed up calculations, logarithms vastly reduced the time required for multiplying numbers with many digits.

They were basic in numerical work for more than 300 years, until the perfection of mechanical calculating machines in the late 19th century and computers in the 20th century rendered them obsolete for large-scale computations. The natural logarithm (with base e ≅ 2.71828 and written ln n), however, continues to be one of the most useful functions in mathematics, with applications to mathematical models throughout the physical and biological sciences.

How to convert the logarithmic form into exponential form?

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g. since 1000 = 10 × 10 × 10 = 103, the "logarithm base 10" of 1000 is 3, or log10 (1000) = 3.

The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation. The logarithm base 10 (that is b = 10) is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e (that is b ≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative. The binary logarithm uses base 2 (that is b = 2) and is frequently used in computer science.

Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors:

{displaystyle log _{b}(xy)=log _{b}x+log _{b}y,}{displaystyle log _{b}(xy)=log _{b}x+log _{b}y,} provided that b, x and y are all positive and b ≠ 1. The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter e as the base of natural logarithms.

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How to solve logarithms without a calculator

Well, to calculate, or i should say to approximate logarithm(base 10) of any number you just need to remember 3 values. And the below method will gurantee correct result upto 2 decimal points.  log 2 = 0.3, log 3 = 0.47 and log e = 0.43.

note, log 5 is log (10/2) i.e log 10- log2 = 0.7

Now, log(1+x) = log e * ln(1+x) .

ln(1+x) approximates to x if |x| << 1

Hence, log(1+x) is approximately equal to 0.43x.

Examples

log 7 = log 6*(1+ 1/6) = log 2+ log 3+ log(1+ 1/6) = 0.3+0.47+ 0.43/6 = 0.84.

log 17 = log 16*(1 + 1/16) = 4log 2 + 0.43/16 = 1.227.

log 11 = log 10*(1 + 1/10) = 1 +0.43/10 =1.043.

log 26 = log 25*(1+ 1/25) = 2log 5+ log (1+ 1/25) =2*0.7 + 0.43/25 = 1.417.

The accuracy can be improved by keeping upto square term in log(1+x), but that would considerably increase computation time.

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Combining product rule and quotient rule in logarithms

log [A cdot B div C div D cdot E] = log;A + log;B - log;C - log;D + log;Elog[A?B÷C÷D?E]=logA+logB?logC?logD+logE)

The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule. Notice that we used the product rule for logarithms to find a solution for the example above.

By doing so, we have derived the power rule for logarithms, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power.

Evaluating logarithms using logarithm rules

We can use the logarithm properties to rewrite logarithmic expressions in equivalent forms.

For example, we can use the product rule to rewrite log(2x)log(2x)log, left parenthesis, 2, x, right parenthesis as log(2)+log(x)log(2)+log(x)log, left parenthesis, 2, right parenthesis, plus, log, left parenthesis, x, right parenthesis. Because the resulting expression is longer, we call this an expansion.

In another example, we can use the change of base rule to rewrite dfrac{ln(x)}{ln(2)}  ln(2) ln(x) start fraction, natural log, left parenthesis, x, right parenthesis, divided by, natural log, left parenthesis, 2, right parenthesis, end fraction as log_2(x)log (x)log, start base, 2, end base, left parenthesis, x, right parenthesis. Because the resulting expression is shorter, we call this a compression. You can also check our free samples for maths.

Examples of How to Solve Exponential Equations using Logarithms

Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since display style mathrm{log}left(aright)=mathrm{log}left(bright)log(a)=log(b) is equivalent to a = b, we may apply logarithms with the same base on both sides of an exponential equation.