Having the knowledge of Characteristic and Mantissa is a prerequisite here.

Examples in Antilog

NOTE: For simplicity, we may use 'log' in place of 'logarithm' and 'logs' in place of 'logarithms'.

In the following problems, besides the knowledge of Characteristic and Mantissa, we make use of theAlgebra Formulas from
Logarithms.

Example 1 of Antilog

Solve the following problem in Logarithm Tables

Given log 2 = 0.3010, log 3 = 0.4771, find the number of digits in 3^{12} x 2^{8}.

Solution to Example 1 of Logarithm Tables :

Let x = 3^{12} x 2^{8} Taking logarithms to base 10 on both sides, we get log x = log (3^{12} x 2^{8})

We know, log of a product can be written as the sum of the logs of the factors of the product (See Formula 5 in Logarithms; Link given above.)

∴ log x = log (3^{12}) + log (2^{8})

We know, in log of a power, the exponent multiplies the log. (See Formula 7 in Logarithms; Link given above.)

∴ log x = 12 (log 3) + 8 (log 2) = 12(0.4771) + 8(0.3010) = 5.7252 + 2.4080 = 8.1332 ⇒ Characteristic of x = 8 ⇒ x has 9 digits. Number of digits in 3^{12} x 2^{8} = 9. Ans.

Example 2 of Antilog

Solve the following problem in Logarithm Tables

Find the number of zeros between the decimal point and the first significant figure in the value of (0.0504)^{12} given that log 2 = 0.3010, log 3 = 0.4771, log 7 = 0.8451.

Solution to Example 2 of Logarithm Tables :

Let x = (0.0504)^{12} x = {504 x 10^{(-4)}}^{12} = (504)^{12} x 10^{(-4 x 12)} = (504)^{12} x 10^{(-48)}

504 is divisible by 9 [ Since 5 + 0 + 4 = 9 is divisible by 9]. ∴ 504 = 9 x 56 = 9 x 7 x 8 = 3^{2} x 7 x 2^{3} (504)^{12} = (3^{2} x 7 x 2^{3})^{12}= 3^{2 x 12} x 7^{12} x 2^{3 x 12} = 3^{24} x 7^{12} x 2^{36}

∴ x = 3^{24} x 7^{12} x 2^{36} x 10^{(-48)} Taking logarithms to base 10 on both sides, we get log x = log {3^{(24)} x 7^{(12)} x 2^{(36)} x 10^{(-48)} }

We know, log of a product can be written as the sum of the logs of the factors of the product (See Formula 5 in Logarithms; Link given above.)

To make Mantissa positive, -15.5724 is written as -16 + 0.4276 Thus log x = 16.4276⇒ Characteristic of x = 16 ⇒ x has 15 zeros between the decimal point and the first significant figure.

The number of zeros between the decimal point and the first significant figure in the value of (0.0504)^{12} is 15. Ans.

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If log_{10} 2 = 0.3010, show that log_{5} 64 = 2.584

Solution to Example 3 of Logarithm Tables :

Let x = log_{5} 64

We know, log of a quantity to a base can be written as the ratio of log of the quantity and log of the base. (See Formula 8 in Logarithms; Link given above.)

∴ x = (log 64)⁄(log 5) = (log 2^{6})⁄{log (10⁄2)}

Numerator is logarithm of a power:

We know, in log of a power, the exponent multiplies the log. (See Formula 7 in Logarithms; Link given above.)

Denominator is logarithm of a quotient:

We know, log of a quotient can be written as the difference of the logs of the numerator and denominator of the quotient (See Formula 6 in Logarithms; Link given above.)

Given log 2 = 0.3010, find the number of digits in 128^{31}.

Find the number of zeros between the decimal point and the first significant figure in the value of (0.0432)^{10} given that log 2 = 0.3010, log 3 = 0.4771

If log_{10} 3 = 0.4771, show that log_{30} 81 = 1.292

For Answers to the problems in Logarithm Tables, see at the bottom of the page.

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