Question

log2 x + log2 (x + 2) = 3​

Answer 1

Step-by-step explanation:

$log_{2}(x) + log_{2}(x + 2) = 3 \\ \\ log_{2}(x(x + 2)) = 3 log_{2}(2) \\ \\ log_{2}( {x}^{2} + 2x ) = log_{2}( {2}^{3} ) = log_{2}(8) \\ \\ {x}^{2} + 2x = 8 \\ \\ {x}^{2} + 2x - 8 = 0 \\ \\ {x}^{2} + 4x - 2x - 8 = 0 \\ \\ x(x + 4) - 2(x + 4) = 0 \\ \\ (x - 2)(x + 4) = 0 \\ \\ x + 4 \neq0 \: \: as \: x + 2 > 0 \\ \\ so \\ \\ x = 2 \\ \\ formulae \\ \\ log_{a}(x) + log_{a}(y) = log_{a}(xy) \\ \\ log_{a}(a) = 1 \\ \\ m log_{a}(x) = log_{a}( {x}^{m} ) \\ \\ if \: \: log_{a}(x) = log_{a}(y) \\ \\ then \: \: x = y \\ \\ and \\ \\ a > 1 \: \: for \: all$

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Answer 2

Answer:

x = 10, or x = e

Step-by-step explanation:

log2x + log2(x+2) = 3

=> log2 + logx + log2x + log 4 = 3

=> 0.301 + logx + log2 + logx + 2log2 = 3

=> 0.301 + logx +0.301 + logx + 0.602 = 3

=> 2logx + 1.204 = 3

=> 2logx = 1.796

=> logx = 0.898

Now, we can assume, 0.898 is nearly equal to 1

=> log x = 1

=> x = 10¹ [ if log base is 10]

=> x = 10

Now, if log base is e, then,

=> log x = 1

=> x = e¹

=> x = e

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