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Simplify the logarithmic expression: 2 4 8

Mathematics

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Answer

3

Step-by-step solution

3 steps

  1. Apply the power rule of logarithms, which states that alog⁡bx=log⁡bxaa \log_b x = \log_b x^a.

    2log⁡48=log⁡4(82)2 \log_4 8 = \log_4 (8^2)

  2. Calculate 828^2.

    82=648^2 = 64

    So the expression becomes:

    log⁡464\log_4 64

  3. Evaluate log⁡464\log_4 64.

    This means finding the power to which 4 must be raised to get 64.
    Let y=log⁡464y = \log_4 64.
    This can be written in exponential form as 4y=644^y = 64.
    We know that 4×4=164 \times 4 = 16, and 16×4=6416 \times 4 = 64.
    So, 43=644^3 = 64.
    Therefore, y=3y = 3.

    The value of the expression is:

    33
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