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Let m be the minimum possible value of log3 ( 3y1 + 3y2+ 3y3) , where y1, y2, y3 are real numbers for which y1 + y2 + y3 = 9. Let M be the maximum possible value of (log3x1 + log3x2 + log3x3), where x1, x2 , x3 are positive real numbers for which x1 + x2 + x3 = 9. Then the value of log2 (m3) + log3(M2) is ______.
Correct answer is '8.00'. Can you explain this answer?
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Let m be the minimum possible value of log3 ( 3y1 + 3y2+ 3y3) , where ...
Thus, log2 (m2) + log3 (m2) = 6 + 2 = 8
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Let m be the minimum possible value of log3 ( 3y1 + 3y2+ 3y3) , where ...
To find the minimum possible value of log3(3y1 3y2 3y3), we need to understand the properties of logarithms.
Firstly, we can rewrite the expression as log3(3y1) + log3(3y2) + log3(3y3).
Using the property of logarithms that states loga(b * c) = loga(b) + loga(c), we can simplify the expression to y1 * log3(3) + y2 * log3(3) + y3 * log3(3).
Since log3(3) = 1, the expression becomes y1 + y2 + y3.
Given that y1 + y2 + y3 = 9, the minimum possible value of the expression is 9.
Therefore, m = 9.
To find the maximum possible value of (log3x1 log3x2 log3x3), we can use a similar approach.
Rewriting the expression, we have log3(x1) + log3(x2) + log3(x3).
Using the property of logarithms loga(b * c) = loga(b) + loga(c), we can simplify the expression to log3(x1 * x2 * x3).
Since x1 * x2 * x3 = 9, the expression becomes log3(9).
Using the property of logarithms loga(b^c) = c * loga(b), we can rewrite the expression as 2 * log3(3).
Since log3(3) = 1, the expression simplifies to 2.
Therefore, M = 2.
To find the value of log2(m^3) log3(M^2), we substitute the values of m and M.
log2(m^3) = log2(9^3) = log2(729) = 9
log3(M^2) = log3(2^2) = log3(4) = 2
Therefore, the value of log2(m^3) log3(M^2) is 9 2 = 8.00.
Firstly, we can rewrite the expression as log3(3y1) + log3(3y2) + log3(3y3).
Using the property of logarithms that states loga(b * c) = loga(b) + loga(c), we can simplify the expression to y1 * log3(3) + y2 * log3(3) + y3 * log3(3).
Since log3(3) = 1, the expression becomes y1 + y2 + y3.
Given that y1 + y2 + y3 = 9, the minimum possible value of the expression is 9.
Therefore, m = 9.
To find the maximum possible value of (log3x1 log3x2 log3x3), we can use a similar approach.
Rewriting the expression, we have log3(x1) + log3(x2) + log3(x3).
Using the property of logarithms loga(b * c) = loga(b) + loga(c), we can simplify the expression to log3(x1 * x2 * x3).
Since x1 * x2 * x3 = 9, the expression becomes log3(9).
Using the property of logarithms loga(b^c) = c * loga(b), we can rewrite the expression as 2 * log3(3).
Since log3(3) = 1, the expression simplifies to 2.
Therefore, M = 2.
To find the value of log2(m^3) log3(M^2), we substitute the values of m and M.
log2(m^3) = log2(9^3) = log2(729) = 9
log3(M^2) = log3(2^2) = log3(4) = 2
Therefore, the value of log2(m^3) log3(M^2) is 9 2 = 8.00.
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Let m be the minimum possible value of log3 ( 3y1 + 3y2+ 3y3) , where y1, y2, y3 are real numbers for which y1 + y2 + y3 = 9. Let M be the maximum possible value of (log3x1 + log3x2 + log3x3), where x1, x2 , x3 are positive real numbers for which x1 + x2 + x3 = 9. Then the value of log2 (m3) + log3(M2) is ______.Correct answer is '8.00'. Can you explain this answer?
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