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Deconstructing the Logarithmic Table for Better Understanding

In mathematics, the logarithm is the most accessible way to express large figures. The logarithm is the power to which each integer has to be raised to gain certain values. The inverse operation of logarithms is also referred to as exponentiation. 

This composition will pass over logarithm features, parcels of logarithmic functions, log value table, log values from 1 to ten for log base 10, and log values from 1 to ten for log base(e).

Having stated that, it allows first to understand with notes what log value is, log functions, how it is represented, its properties, and how it affects our computations.

Log features

The logarithm function is the inverse of the exponentiation feature. The logarithms function is defined as follows:

F (x) = loga (x)

The base of the logarithm is a, in this situation. It’s written as a log base ofx. Base 10 and base e are the most frequently used logarithm functions.

Logarithm function types

Common Logarithms function-The logarithm function with base 10 is referred to as the common Logarithms function. It is written as log10.

F (x) = log10 (x)

Natural Logarithms function – Natural Logarithms characteristic is the logarithm function with base. It’s written as loge.

F (x) = loge (x)

Properties of Logarithm functions

Product rule: The product rule involves multiplying figures with the same base and additionally including the exponents.

Logb (MN) = Logb (M) Logb (N)

Quotient Rule: The antilog of the difference of logarithms of the 2 figures is the department.

In other words, the difference of the logarithms of two figures, say a and b, is the same as the logarithm of their division. Each figure has to have an equal base. That is known as the quotient rule for logarithms.

Logb (M/ N) = Logb (M)-Logb (N)

Power Rule:  Exponents’ expressions are raised to power in the power rule, and also the exponents are multiplied.

Logb (Mp) = P logb (M)

Zero Exponent rules

Loga = 1

Change of Base Rule

Logb (x) = in x/ In b or logb (x) = log10 x / log10 b

Value of Log 1 to log 10 for Log Base 10 table

Right here, we can list the log values from 1 to 10 for log base 10 in tabular format.

Log Table 1 to 10 for Log Base 10

Common Logarithm to a Number (log10 x) Log Values
Log 1 0
Log 2 0.3010
Log 3 0.4771
Log 4 0.6020
Log 5 0.6989
Log 6 0.7781
Log 7 0.8450
Log 8 0.9030
Log 9 0.9542
Log 10 1

 

Log Table 1 to 10 for Log Base e

Common logarithm to a Number (loge x) Ln Value​
In (1) 0
In (2) 0.693147
In (3) 1.098612
In (4) 1.386294
In (5) 1.609438
In (6) 1.791759
In (7) 1.94591
In (8) 2.079442
In (9) 2.197225
In (10) 2.302585

A few important questions that you’ll encounter are as follows:

Question 1: What’s the best way to read a log table?

Regardless of the decimal, take the first two integers of the number and look for the row. Also, check for the column number that corresponds to the number’s third number. To get the final figure, you may need to consult the mean difference table.

Question 2: What’s the value of log 1 to 10?

A Log 1 to the base 10 is equal to 0.

In certain ways, logs can be thought of as exponents. Isn’t it perplexing? To further grasp this, let’s look at some examples.

Using the log10 method (“log to the base 10”)

102 = 100 is similar to log10(100) = 2.

where 10 is the base, 2 is the logarithm (or power), and 100 is the number

Calculus was used to derive several chemistry equations, many of which utilised natural logarithms. Why is ln x = 2.303 log x the relationship between ln x and log x? Let’s take a look at x = 10 and see what we can find out.

We have (ln 10)/ (log 10) = number after rearranging.

ln 10 = 2.302585093. or 2.303, and log 10 = 1.

So, the number has to be 2.303. Voila!

Using natural logs ( loge or ln)

Carrying all figures to 5 significant numbers,

ln 30 = 3.4012 is equivalent to e3.4012 = 30 or2.71833.4012 = 30

Now let’s find some further log examples but take a number that has power other than 10. You will require your scientific calculator in this case. Alternately, you can use the log table as well, whichever is available. On most calculators, you gain the log (or ln) of a number by-

First enter the number, followed by,

Clicking the log (or ln) button.

Example 1 log5.43 x 1010 = 10.73479983.

Example 2 log2.7 x 10-8 = -7.568636236.

The decimal portion is now too large in both circumstances. So, what are our options for dealing with this?

Let’s look at the logarithm closely to see how we might determine how many significant numbers it should have.

For each log, the mantissa is the number to the right of the decimal point, and the characteristic is the number to the left of the decimal point. When computing the number of significant numbers, the characteristic is often ignored because it just locates the value’s decimal point. The amount of significant numbers in the mantissa is the same as the number of logs detected. Consider the following scenario as an example:

Example 1 log5.43 x 1010 = 10.735

The number has 3 significant numbers, but its log ends with 5 since the mantissa has 3 and the character has 2.

Example 2 log2.7 x 10-8 = -7.57

The number has 2 significant numbers, but its log ends up with 3 significant numbers.

Natural logarithms work in the same way.

Example 3 ln3.95 x 106 = 15.18922614. = 15.189

Before we wrap this up, let’s touch base on one last point, inverse log.

Researching For Antilogarithms (also Called Inverse Logarithm)

We sometimes know the logarithm (or ln) but work backwards to find the number itself. Finding a number’s antilogarithm or inverse logarithm is known as finding the antilogarithm or inverse logarithm. You’ll need one of the most basic scientific calculators you can find to do so.

Press the inverse (inv) or shift button, as well as the log (or ln) button, after entering the value. It’s also known as the 10x (or ex) button.

Example 5 log x = 4.203; so, x = inverse log of4.203 = 15958.79147

There are three significant numbers in the mantissa of the log, so the number has 3 significant numbers. The answer to the correct number of significant numbers is1.60 x 104.

Example 6 log x = -15.3;

so, x = inv log (-15.3) = 5.011872336. x 10-16 = 5 x 10-16 (1 significant figure)

Conclusion

Therefore, we hope you now understand log values, their functions and their different properties. 

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