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Logarithms - Understanding Logarithms through powers of 10-advanced

Grade 9CBSE

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

🔑Concepts

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The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. For common logarithms, the base is 1010.

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If 10x=y10^x = y, then log⁡10(y)=x\log_{10}(y) = x. This is the fundamental relationship between exponents and logarithms.

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For powers of 1010 greater than 11, the logarithm is positive and equal to the number of zeros after 11. For example, log⁡10(100)=2\log_{10}(100) = 2 because 102=10010^2 = 100.

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The logarithm of 11 is always 00 regardless of the base, because 100=110^0 = 1. Thus, log⁡10(1)=0\log_{10}(1) = 0.

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For decimal values between 00 and 11 that are powers of 1010, the logarithm is negative. For example, log⁡10(0.1)=−1\log_{10}(0.1) = -1 because 10−1=11010^{-1} = \frac{1}{10}.

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Advanced property: The logarithm of a number in the form 10n×10m10^n \times 10^m can be found using the addition law: log⁡10(10n×10m)=n+m\log_{10}(10^n \times 10^m) = n + m.

📐Formulae

log⁡10(10n)=n\log_{10}(10^n) = n

log⁡10(1)=0\log_{10}(1) = 0

log⁡10(10)=1\log_{10}(10) = 1

log⁡10(110n)=−n\log_{10}\left(\frac{1}{10^n}\right) = -n

log⁡10(mn)=nlog⁡10(m)\log_{10}(m^n) = n \log_{10}(m), where m=10m = 10

log⁡10(10n)=1n\log_{10}(\sqrt[n]{10}) = \frac{1}{n}

💡Examples

Problem 1:

Evaluate the value of log⁡10(1,00,000)\log_{10}(1,00,000).

Solution:

We know that 1,00,0001,00,000 can be written as a power of 1010. 1,00,000=1051,00,000 = 10^5 Using the identity log⁡10(10n)=n\log_{10}(10^n) = n: log⁡10(105)=5\log_{10}(10^5) = 5

Explanation:

Since there are five zeros in 1,00,0001,00,000, the exponent required for base 1010 is 55.

Problem 2:

Find the value of xx if log⁡10(x)=−4\log_{10}(x) = -4.

Solution:

By converting the logarithmic equation to exponential form: x=10−4x = 10^{-4} x=1104x = \frac{1}{10^4} x=110000x = \frac{1}{10000} x=0.0001x = 0.0001

Explanation:

A negative logarithm indicates that the number is a fraction (decimal) less than 11 but greater than 00.

Problem 3:

Evaluate log⁡10(10010)\log_{10}(100 \sqrt{10}).

Solution:

First, express the term inside the logarithm as a single power of 1010: 10010=102×1012100 \sqrt{10} = 10^2 \times 10^{\frac{1}{2}} Using the law of exponents am×an=am+na^m \times a^n = a^{m+n}: 102+12=105210^{2 + \frac{1}{2}} = 10^{\frac{5}{2}} Now, apply the logarithm: log⁡10(1052)=52=2.5\log_{10}(10^{\frac{5}{2}}) = \frac{5}{2} = 2.5

Explanation:

We combine the powers of 1010 using exponent rules before applying the logarithmic definition.