Answer :

To evaluate [tex]\log_5 4 + \log_2^{-1} 125[/tex], let's break it down step-by-step:


  1. Evaluate [tex]\log_5 4[/tex]:

    Logarithms represent the power to which a base number must be raised to equal another number. Here, [tex]\log_5 4[/tex] asks: "To what power must 5 be raised to get 4?" This is not a simple integer, and without a calculator, we can't simplify this further directly. For the purpose of solving, you may need a calculator or logarithm table to approximate this value.


  2. Evaluate [tex]\log_2^{-1} 125[/tex]:

    First, simplify [tex]\log_2^{-1} 125[/tex] to understand it better:


    • [tex]\log_2^{-1} 125[/tex] is equivalent to [tex]- \log_2 125[/tex], because [tex]x^{-1} = \frac{1}{x}[/tex] and [tex]\log_b \left( \frac{1}{x} \right) = -\log_b x[/tex].

    • So, this becomes [tex]-\log_2 125[/tex].


    To find [tex]\log_2 125[/tex], think of it as: "To what power must 2 be raised to get 125?" Again, this isn't a simple integer, and you will typically use a calculator for an approximate decimal result.


  3. Combine the values:

    The expression becomes:
    [tex]\log_5 4 - \log_2 125[/tex]

    Since the values are not easily calculable without computational tools or graphs, the expression can be approximated using a calculator.



Remember, without computational aid, this expression does not resolve to a neat, simple answer. However, in practical scenarios, tools like calculators are utilized to find decimal approximations of such logarithmic expressions.

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