Answer :

To convert the equation [tex]\(197,826.9064 = 10^w\)[/tex] into logarithmic form, follow these steps:

1. Identify the Base of the Exponent:
The given equation is in the form of [tex]\(b^x = y\)[/tex], where [tex]\(b\)[/tex] is the base of the exponent. In this case, [tex]\(b\)[/tex] is 10.

2. Logarithmic Form Definition:
The logarithmic form of the equation [tex]\(b^x = y\)[/tex] is [tex]\(\log_b(y) = x\)[/tex]. This means that the logarithm base [tex]\(b\)[/tex] of [tex]\(y\)[/tex] equals [tex]\(x\)[/tex].

3. Apply to the Given Equation:
For the equation [tex]\(197,826.9064 = 10^w\)[/tex], the base [tex]\(b\)[/tex] is 10, and the number [tex]\(y\)[/tex] is 197,826.9064. We want to find the exponent [tex]\(w\)[/tex] such that:

[tex]\[
\log_{10}(197,826.9064) = w
\][/tex]

4. Result:
Thus, the equation [tex]\(197,826.9064 = 10^w\)[/tex] is equivalent to its logarithmic form [tex]\(\log_{10}(197,826.9064) = w\)[/tex].

5. Calculate the Logarithmic Value:
The value of [tex]\(w\)[/tex] is approximately 5.296285359588532.

So, the logarithmic form of the given equation is [tex]\(\log_{10}(197,826.9064) \approx 5.296285359588532\)[/tex].

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