Log_2 64 + log_2 16 - log_2 9

Mathematics

ScanSolve AI Answer

Answer

10−log⁡2910 - \log_2 9

Step-by-step solution

3 steps

To solve the expression log⁡264+log⁡216−log⁡29\log_2 64 + \log_2 16 - \log_2 9, we will evaluate each logarithmic term.

  1. Evaluate log⁡264\log_2 64.

    We need to find the power to which 2 must be raised to get 64.

    2x=642^{x} = 64

    Since 26=642^{6} = 64, we have log⁡264=6\log_2 64 = 6.

  2. Evaluate log⁡216\log_2 16.

    We need to find the power to which 2 must be raised to get 16.

    2y=162^{y} = 16

    Since 24=162^{4} = 16, we have log⁡216=4\log_2 16 = 4.

  3. Substitute the evaluated terms into the original expression.

    The term log⁡29\log_2 9 cannot be simplified to an integer, as 9 is not an integer power of 2.
    Substitute the values from Step 1 and Step 2 into the expression:

    log⁡264+log⁡216−log⁡29=6+4−log⁡29\log_2 64 + \log_2 16 - \log_2 9 = 6 + 4 - \log_2 9 6+4−log⁡29=10−log⁡296 + 4 - \log_2 9 = 10 - \log_2 9

    The final answer is 10−log⁡2910 - \log_2 9.

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