For non-negative integers, 𝑎, 𝑏, 𝑐, what would be the value of 𝑎 + 𝑏 + 𝑐 if log 𝑎 + log 𝑏 + log 𝑐 = 0?
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Logarithm Problem: Finding Sum of Non-Negative Integers
This problem asks us to find the value of \(a + b + c\) given that \(a\), \(b\), and \(c\) are non-negative integers, and their logarithms sum to zero. The given equation is \(\log a + \log b + \log c = 0\).
Logarithm Properties Application
To solve this, we first need to use a fundamental property of logarithms. The sum of logarithms with the same base is equal to the logarithm of the product of their arguments. This property is expressed as:
- \(\log_B M + \log_B N = \log_B (MN)\)
Applying this property to our given equation \(\log a + \log b + \log c = 0\), we can combine the terms on the left side:
\(\log (a \cdot b \cdot c) = 0\)
Determining the Product of Integers (abc)
Now, we need to consider what it means for a logarithm to be equal to zero. For any valid logarithm base \(B\) (where \(B > 0\) and \(B \neq 1\)), if \(\log_B X = 0\), then the argument \(X\) must be equal to 1. This is because any non-zero number raised to the power of 0 is 1 (i.e., \(B^0 = 1\)).
Therefore, from the equation \(\log (a \cdot b \cdot c) = 0\), we can conclude that the product of \(a\), \(b\), and \(c\) must be 1:
\(a \cdot b \cdot c = 1\)
Finding the Values of a, b, c
We are given that \(a\), \(b\), and \(c\) are non-negative integers. This means they can be \(0, 1, 2, 3, \ldots\). We need to find values for \(a\), \(b\), and \(c\) that satisfy both conditions: they are non-negative integers, and their product is 1.
Let's analyze the possibilities:
- If any of \(a\), \(b\), or \(c\) were \(0\), their product \(a \cdot b \cdot c\) would be \(0\), not \(1\). Thus, \(a, b, c\) cannot be \(0\).
- Since they must be integers and their product is \(1\), the only positive integer solution is if each integer itself is \(1\).
Therefore, the only possible integer values for \(a\), \(b\), and \(c\) that satisfy \(a \cdot b \cdot c = 1\) and are non-negative are:
- \(a = 1\)
- \(b = 1\)
- \(c = 1\)
These values also ensure that the original logarithm expression \(\log a + \log b + \log c\) is defined, as logarithms are defined only for positive arguments (\(X > 0\)). Since \(1 > 0\), \(\log 1\) is defined (and is equal to 0).
Calculating the Sum \(a + b + c\)
Now that we have determined the individual values of \(a\), \(b\), and \(c\), we can find their sum:
\(a + b + c = 1 + 1 + 1\)
\(a + b + c = 3\)
Thus, the value of \(a + b + c\) is 3.
Important Questions from Verbal Deduction
Lamenting the gradual sidelining of the arts in school curricula, a goup of prominent artists wrote to the Chief Minister last year, asking him to allocate more funds to support arts education in schools. However, no such increase has been announced in this year`s Budget. The artists expressed their deep anguish at their request not being approved, but many of them remain optimistic about funding in the future.
Which of the statement(s) below is/are logically valid and can be inferred from the above statements?
(i) The artists expected funding for the arts to increase this year.
(ii) The Chief Minister was receptive to the idea of increasing funding for the arts.
(iii) The Chief Minister is a prominent artist.
(iv) Schools are giving less importance to arts education nowadays,Read the following paragraph and choose the correct statement.
Climate change has reduced human security and threatened human wellbeing. An ignored reality of human progress is that human security largely depends upon environmental security. But on the contrary, human progress seems contradictory to environmental security. To keep up both at the required level is a challenge to be addressed by one and all. One of the ways to curb the climate change may be suitable scientific innovations, while the other may be the Gandhian perspective on
Small scale progress with focus on sustainability.Choose the most appropriate phrase from the options given below to complete the following sentence.
The aircraft_______ take off as soon as its flight plan was filed.
Once the team of analysts identifies the problem, we _________ in a better position to comment on the issue.
Which one of the following choices CANNOT fill the given blank?“If you are looking for a history of India, or for an account of the rise and fall of the British Raj, or for the reason of the cleaving the subcontinent into two mutually antagonistic parts and the effects this mutilation will have in the respective sections, and ultimately on Asia, you will to find it in these pages; for though I have spent a lifetime in the country, I lived too near the seat of events, and was too intimately associated with the actors, to get the perspective needed for the impartial recording of these matters.”
Which of the following is closest in meaning to ‘cleaving’?
