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Loga^b logb^c = 0 then?
Most Upvoted Answer
Loga^b logb^c = 0 then?
Explanation:
The given equation is:
logab × logbc = 0
Rule of logarithms:
If logab = x, then ax = b
Using the above rule of logarithms:
logab × logbc = 0
If logab = x, then ax = b
If logbc = y, then by = c
Therefore, we can rewrite the given equation as:
ax × by = 1
Since ax and by are both positive, their product can only be equal to 1 if both ax and by are equal to 1.
Using the above conclusion:
Therefore, we have:
ax = 1 and by = 1
If ax = 1, then x = 0 (since a0 = 1 for any positive value of a)
If by = 1, then y = 0 (since b0 = 1 for any positive value of b)
Therefore, we have:
logab = 0 and logbc = 0
If logab = 0, then b = a0 = 1
If logbc = 0, then c = b0 = 1
Therefore, we have:
b = 1 and c = 1
Substituting these values in the original equation, we get:
loga1 × log11 = 0
Since loga1 = 0 (for any positive value of a) and log11 = 0, we have:
0 × 0 = 0
Conclusion:
Therefore, the given equation logab × logbc = 0 is true if and only if b = 1 and c = 1.
Community Answer
Loga^b logb^c = 0 then?
If the equation is loga^b * logb^c = 0, we can simplify it using the properties of logarithms.
First, let's recall the following properties:
loga^b = c if and only if a^c = b.
loga^b * logb^c = loga^b^c.
Using property 1, we can rewrite the equation as:
(a^c) * (b^c) = 1.
Now, let's analyze the possibilities:
If a^c = 1 and b^c = 1, then the equation is satisfied.
If a^c ≠ 1 and b^c = 1, then the equation is not satisfied because (a^c) * (b^c) ≠ 1.
If a^c = 1 and b^c ≠ 1, then the equation is not satisfied because (a^c) * (b^c) ≠ 1.
If a^c ≠ 1 and b^c ≠ 1, then the equation is not satisfied because (a^c) * (b^c) ≠ 1.
Therefore, the only possibility for the equation loga^b * logb^c = 0 to be true is if a^c = 1 and b^c = 1.
First, let's recall the following properties:
loga^b = c if and only if a^c = b.
loga^b * logb^c = loga^b^c.
Using property 1, we can rewrite the equation as:
(a^c) * (b^c) = 1.
Now, let's analyze the possibilities:
If a^c = 1 and b^c = 1, then the equation is satisfied.
If a^c ≠ 1 and b^c = 1, then the equation is not satisfied because (a^c) * (b^c) ≠ 1.
If a^c = 1 and b^c ≠ 1, then the equation is not satisfied because (a^c) * (b^c) ≠ 1.
If a^c ≠ 1 and b^c ≠ 1, then the equation is not satisfied because (a^c) * (b^c) ≠ 1.
Therefore, the only possibility for the equation loga^b * logb^c = 0 to be true is if a^c = 1 and b^c = 1.
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Loga^b logb^c = 0 then?
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Loga^b logb^c = 0 then? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about Loga^b logb^c = 0 then? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Loga^b logb^c = 0 then?.
Loga^b logb^c = 0 then? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about Loga^b logb^c = 0 then? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Loga^b logb^c = 0 then?.
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