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15 May, 09:00

A teacher used the change of base formula to determine whether the equation below is correct.

(log Subscript 2 Baseline 10) (log Subscript 4 Baseline 8) (log Subscript 10 Baseline 4) = 3

Which statement explains whether the equation is correct?

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Answers (1)
  1. 15 May, 10:39
    0
    The equation is correct

    Step-by-step explanation:

    The equation, written as:

    [log_2 (10) ][log_4 (8) ][log_10 (4) ] = 3

    Consider the change of base formula:

    log_a (x) = [log_10 (x) ] / [log_10 (a) ]

    Applying the change of base formula to change the expressions in base 2 and base 4 to base 10.

    (1)

    log_2 (10) = [log_10 (10) ]/[log_10 (2) ]

    = 1/[log_10 (2) ]

    (Because log_10 (10) = 1)

    (2)

    log_4 (8) = [log_10 (8) ]/[log_10 (4) ]

    Now putting the values of these new logs in base 10 into the left-hand side of original equation to verify if we have 3, we have:

    [log_10 (2) ][log_8 (4) ][log_10 (4) ]

    = [1 / log_10 (2) ][log_10 (8) / log_10 (4) ][log_10 (4) ]

    = [1/log_10 (2) ] [log_10 (8) ]

    = [log_10 (8) ]/[log_10 (2) ]

    = [log_10 (2³) ]/[log_10 (2) ]

    Since log_b (a^x) = xlog_b (a)

    = 3[log_10 (2) ]/[log_10 (2) ]

    = 3 as required

    Therefore, the left hand side of the equation is equal to the right hand side of the equation.
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