Question

write the expression as a single logarithm using the properties of logarthms
3 log x + 2 log 5 + log y

Answer 1

To combine the expression \(3 \log x + 2 \log 5 + \log y\) into a single logarithm, we can use the properties of logarithms.

1. **Power Rule**: \(a \log b = \log(b^a)\)
2. **Product Rule**: \(\log a + \log b = \log(ab)\)

Let's apply these rules step by step:

1. Start with \(3 \log x\):
\[
3 \log x = \log(x^3)
\]

2. Next, apply the power rule to \(2 \log 5\):
\[
2 \log 5 = \log(5^2) = \log(25)
\]

3. Now we can rewrite the entire expression using these results:
\[
3 \log x + 2 \log 5 + \log y = \log(x^3) + \log(25) + \log(y)
\]

4. Finally, apply the product rule:
\[
\log(x^3) + \log(25) + \log(y) = \log(x^3 \cdot 25 \cdot y)
\]

Thus, the entire expression simplifies to:
\[
\log(25x^3y)
\]

So the expression \(3 \log x + 2 \log 5 + \log y\) as a single logarithm is:
\[
\log(25x^3y)
\]