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Logarithms - Logarithm to base 10-advanced

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

πŸ”‘Concepts

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Common Logarithms: Logarithms with base 1010 are known as common logarithms. If no base is mentioned, it is usually assumed to be 1010 in numerical calculations.

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Standard Form: Any positive number NN can be expressed in the form N=mΓ—10nN = m \times 10^n, where 1≀m<101 \le m < 10 and nn is an integer.

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Characteristic and Mantissa: The logarithm of a number consists of two parts. The integral part is called the characteristic and the decimal (fractional) part is called the mantissa.

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Rules for Characteristic: 1. If N>1N > 1, the characteristic is nβˆ’1n-1, where nn is the number of digits to the left of the decimal point. 2. If 0<N<10 < N < 1, the characteristic is negative and is given by βˆ’(k+1)-(k+1), where kk is the number of zeros immediately after the decimal point. This is often written with a bar, e.g., 1Λ‰\bar{1}, 2Λ‰\bar{2}.

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Mantissa Property: The mantissa of the log of all numbers having the same sequence of digits is the same. The mantissa is always kept positive.

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Number of Digits: To find the number of digits in xnx^n, calculate y=log⁑10(xn)y = \log_{10}(x^n). If the characteristic is CC, the number of digits is C+1C + 1.

πŸ“Formulae

log⁑10(MΓ—N)=log⁑10M+log⁑10N\log_{10} (M \times N) = \log_{10} M + \log_{10} N

log⁑10(MN)=log⁑10Mβˆ’log⁑10N\log_{10} \left(\frac{M}{N}\right) = \log_{10} M - \log_{10} N

log⁑10(Mk)=klog⁑10M\log_{10} (M^k) = k \log_{10} M

log⁑ab=log⁑10blog⁑10a\log_{a} b = \frac{\log_{10} b}{\log_{10} a}

log⁑1010=1, log⁑101=0\log_{10} 10 = 1, \text{ } \log_{10} 1 = 0

NumberΒ N=10Characteristic+Mantissa\text{Number } N = 10^{\text{Characteristic} + \text{Mantissa}}

πŸ’‘Examples

Problem 1:

Find the number of digits in 3503^{50}, given that log⁑103=0.4771\log_{10} 3 = 0.4771.

Solution:

Let x=350x = 3^{50}. Taking log on both sides: log⁑10x=log⁑10(350)\log_{10} x = \log_{10} (3^{50}) log⁑10x=50Γ—log⁑103\log_{10} x = 50 \times \log_{10} 3 log⁑10x=50Γ—0.4771\log_{10} x = 50 \times 0.4771 log⁑10x=23.855\log_{10} x = 23.855 Here, the characteristic is 2323. Number of digits =Characteristic+1=23+1=24= \text{Characteristic} + 1 = 23 + 1 = 24.

Explanation:

To find the number of digits in an exponential expression, we find its common logarithm. The integral part (characteristic) plus one gives the total number of digits.

Problem 2:

Given log⁑102=0.3010\log_{10} 2 = 0.3010 and log⁑103=0.4771\log_{10} 3 = 0.4771, find the value of log⁑1012\log_{10} 12.

Solution:

log⁑1012=log⁑10(22Γ—3)\log_{10} 12 = \log_{10} (2^2 \times 3) log⁑1012=log⁑10(22)+log⁑103\log_{10} 12 = \log_{10} (2^2) + \log_{10} 3 log⁑1012=2log⁑102+log⁑103\log_{10} 12 = 2 \log_{10} 2 + \log_{10} 3 Substituting the values: log⁑1012=2(0.3010)+0.4771\log_{10} 12 = 2(0.3010) + 0.4771 log⁑1012=0.6020+0.4771\log_{10} 12 = 0.6020 + 0.4771 log⁑1012=1.0791\log_{10} 12 = 1.0791

Explanation:

We express 1212 in terms of its prime factors 22 and 33, then use the product and power laws of logarithms to substitute the given values.

Problem 3:

If log⁑10x=2Λ‰.3010\log_{10} x = \bar{2}.3010, find the characteristic and mantissa.

Solution:

The given logarithm is log⁑10x=2Λ‰.3010\log_{10} x = \bar{2}.3010. Characteristic =βˆ’2= -2 (represented as 2Λ‰\bar{2}). Mantissa =0.3010= 0.3010. Note that 2Λ‰.3010\bar{2}.3010 actually means βˆ’2+0.3010=βˆ’1.6990-2 + 0.3010 = -1.6990.

Explanation:

In bar notation, only the integral part is negative while the mantissa remains positive. This facilitates the use of log tables.