Question

Practice

Answer 1

Let's expand and simplify each of the logarithmic expressions one by one.

1. **log₃(8x²y)**
Using the properties of logarithms:
\[
\log₃(8x²y) = \log₃(8) + \log₃(x²) + \log₃(y)
\]
Further expanding:
\[
\log₃(8) = \log₃(2³) = 3\log₃(2)
\]
and,
\[
\log₃(x²) = 2\log₃(x)
\]
Therefore, the complete expansion is:
\[
\log₃(8x²y) = 3\log₃(2) + 2\log₃(x) + \log₃(y)
\]

2. **log((x/9)⁵)**
Using the power and quotient rules:
\[
\log((x/9)⁵) = 5\log(x/9) = 5(\log(x) - \log(9))
\]
Further expanding \(\log(9)\):
\[
\log(9) = \log(3²) = 2\log(3)
\]
Therefore, the final expression is:
\[
\log((x/9)⁵) = 5\log(x) - 10\log(3)
\]

3. **log_b(b/x)**
Using the properties of logarithms:
\[
\log_b(b/x) = \log_b(b) - \log_b(x)
\]
Since \(\log_b(b) = 1\), we have:
\[
\log_b(b/x) = 1 - \log_b(x)
\]

4. **log(x²y³/z⁴)**
Using the properties of logarithms:
\[
\log(x²y³/z⁴) = \log(x²y³) - \log(z⁴)
\]
Further expanding:
\[
\log(x²y³) = \log(x²) + \log(y³) = 2\log(x) + 3\log(y)
\]
and,
\[
\log(z⁴) = 4\log(z)
\]
Therefore, the complete expansion is:
\[
\log(x²y³/z⁴) = 2\log(x) + 3\log(y) - 4\log(z)
\]

5. **log₄(4√(3x³))**
Recall that \(4\sqrt{(3x³)} = (3x³)^{1/4}\) and \(4 = 4^{1}\):
\[
\log₄(4√(3x³)) = \log₄((3x³)^{1/4}) + \log₄(4)
\]
Simplifying it gives:
\[
\log₄(4) = 1
\]
and,
\[
\log₄((3x³)^{1/4}) = \frac{1}{4} \log₄(3x³) = \frac{1}{4} (\log₄(3) + 3\log₄(x))
\]
Thus, the entire expression simplifies to:
\[
\log₄(4√(3x³)) = 1 + \frac{1}{4} \log₄(3) + \frac{3}{4} \log₄(x)
\]

6. **log₂(√(m/n²))**
Using the property of logarithms:
\[
\log₂(√(m/n²)) = \log₂((m/n²)^{1/2}) = \frac{1}{2} \log₂(m/n²)
\]
This expands to:
\[
\frac{1}{2} (\log₂(m) - \log₂(n²)) = \frac{1}{2} (\log₂(m) - 2\log₂(n)) = \frac{1}{2} \log₂(m) - \log₂(n)
\]

Thus, the final expansions and simplifications are as follows:

1. \(\log₃(8x²y) = 3\log₃(2) + 2\log₃(x) + \log₃(y)\)
2. \(\log((x/9)⁵) = 5\log(x) - 10\log(3)\)
3. \(\log_b(b/x) = 1 - \log_b(x)\)
4. \(\log(x²y³/z⁴) = 2\log(x) + 3\log(y) - 4\log(z)\)
5. \(\log₄(4√(3x³)) = 1 + \frac{1}{4}\log₄(3) + \frac{3}{4}\log₄(x)\)
6. \(\log₂(√(m/n²)) = \frac{1}{2}\log₂(m) - \log₂(n)\)