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If loga b + loga c=0 then
- a)b=c
- b)b=-c
- c)b=c=1
- d)b and c are reciprocals.
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If logab + logac=0 thena)b=cb)b=-cc)b=c=1d)b and c are reciprocals.Cor...
Given the equation:
loga b + loga c = 0
Step 1: Apply the logarithmic sum property
The sum of two logarithms with the same base can be combined:
loga b + loga c = loga (b ⋅ c)
So the equation becomes:
loga (b ⋅ c) = 0
The sum of two logarithms with the same base can be combined:
loga b + loga c = loga (b ⋅ c)
So the equation becomes:
loga (b ⋅ c) = 0
Step 2: Understand the meaning of loga x = 0
From logarithmic properties, we know that:
loga x = 0 implies x = a0
And since a0 = 1, we get:
x = 1
From logarithmic properties, we know that:
loga x = 0 implies x = a0
And since a0 = 1, we get:
x = 1
Step 3: Apply this to b ⋅ c
From loga (b ⋅ c) = 0, we conclude:
b ⋅ c = 1
Final Result:
If loga b + loga c = 0, then:
b ⋅ c = 1
From loga (b ⋅ c) = 0, we conclude:
b ⋅ c = 1
Final Result:
If loga b + loga c = 0, then:
b ⋅ c = 1
Most Upvoted Answer
If logab + logac=0 thena)b=cb)b=-cc)b=c=1d)b and c are reciprocals.Cor...
Solution:
Given: logab logac=0
To find: Possible values of b and c
Using Laws of Logarithms, we can simplify the given expression as follows:
logab logac=0
=> loga(b × c) / loga(c × c) = 0 (Using product and power rule of logarithms)
=> loga(b × c) / 2 logac = 0 (Using quotient rule of logarithms)
=> loga(b × c) = 0 (Dividing both sides by 2 logac)
=> b × c = a^0 (Using definition of logarithms)
=> b × c = 1
Therefore, we have b × c = 1.
Possible values of b and c:
1. b = c = 1:
If b = c = 1, then b × c = 1 × 1 = 1, which satisfies the given condition.
2. b = 1/c:
If b = 1/c, then b × c = (1/c) × c = 1, which also satisfies the given condition.
Hence, the possible values of b and c are:
b = c = 1 or b = 1/c.
Therefore, the correct option is (c) b = c = 1.
Given: logab logac=0
To find: Possible values of b and c
Using Laws of Logarithms, we can simplify the given expression as follows:
logab logac=0
=> loga(b × c) / loga(c × c) = 0 (Using product and power rule of logarithms)
=> loga(b × c) / 2 logac = 0 (Using quotient rule of logarithms)
=> loga(b × c) = 0 (Dividing both sides by 2 logac)
=> b × c = a^0 (Using definition of logarithms)
=> b × c = 1
Therefore, we have b × c = 1.
Possible values of b and c:
1. b = c = 1:
If b = c = 1, then b × c = 1 × 1 = 1, which satisfies the given condition.
2. b = 1/c:
If b = 1/c, then b × c = (1/c) × c = 1, which also satisfies the given condition.
Hence, the possible values of b and c are:
b = c = 1 or b = 1/c.
Therefore, the correct option is (c) b = c = 1.
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