Question

Use properties of logarithms to expand the logarithmic expression as much as possible. Where​ possible, evaluate logarithmic expressions without using a calculator.
log left parenthesis 100 comma 000 y right parenthesis
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Part 1
log left parenthesis 100 comma 000 y right parenthesisequals
  
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Answer 1

To expand the logarithmic expression \(\log(100,000y)\) using properties of logarithms, we can apply the product rule, which states that \(\log_b(mn) = \log_b(m) + \log_b(n)\).

1. We can first express \(100,000\) in terms of \(10\):
\[
100,000 = 10^5
\]

2. Therefore, we have:
\[
\log(100,000y) = \log(10^5 \cdot y)
\]

3. Now, apply the product rule:
\[
\log(100,000y) = \log(10^5) + \log(y)
\]

4. Next, we can evaluate \(\log(10^5)\) using the power rule of logarithms, which states that \(\log_b(m^n) = n \cdot \log_b(m)\):
\[
\log(10^5) = 5 \cdot \log(10)
\]
Since \(\log(10) = 1\) (if the base is 10), we have:
\[
\log(10^5) = 5 \cdot 1 = 5
\]

5. Putting it all together, the expanded form is:
\[
\log(100,000y) = 5 + \log(y)
\]

So the answer is:
\[
\log(100,000y) = 5 + \log(y)
\]