Question

log6 (5/x)
log2 (z/32)
In (x^3/m)
In (x/4a)
log (a^4/pq)

Answer 1

It looks like you're interested in simplifying or rewriting logarithmic expressions using logarithm properties. Let's go through each of the expressions step by step:

1. **log₆(5/x)**:
Using the property of logarithms that states \(\log_b(a/c) = \log_b(a) - \log_b(c)\):
\[
\log_6(5/x) = \log_6(5) - \log_6(x)
\]

2. **log₂(z/32)**:
Using the same property:
\[
\log_2(z/32) = \log_2(z) - \log_2(32)
\]
Since \(32 = 2^5\), we have:
\[
\log_2(32) = 5
\]
Therefore:
\[
\log_2(z/32) = \log_2(z) - 5
\]

3. **In(x³/m)**:
Assuming "In" represents the natural logarithm (ln), we can apply the same logarithmic property:
\[
\ln(x^3/m) = \ln(x^3) - \ln(m)
\]
Using the power rule \(\ln(a^b) = b \cdot \ln(a)\):
\[
\ln(x^3) = 3 \cdot \ln(x)
\]
Therefore:
\[
\ln(x^3/m) = 3 \cdot \ln(x) - \ln(m)
\]

4. **In(x/4a)**:
Again, using the logarithmic property:
\[
\ln(x/4a) = \ln(x) - \ln(4a) = \ln(x) - (\ln(4) + \ln(a))
\]
Since \(4 = 2^2\), we have:
\[
\ln(4) = 2 \cdot \ln(2)
\]
Thus:
\[
\ln(x/4a) = \ln(x) - 2 \cdot \ln(2) - \ln(a)
\]

5. **log(a⁴/pq)**:
Using the logarithmic property:
\[
\log(a^4/pq) = \log(a^4) - \log(pq)
\]
Applying the properties:
\[
\log(a^4) = 4 \cdot \log(a) \quad \text{and} \quad \log(pq) = \log(p) + \log(q)
\]
Therefore:
\[
\log(a^4/pq) = 4 \cdot \log(a) - (\log(p) + \log(q))
\]
Which can be expanded to:
\[
= 4 \cdot \log(a) - \log(p) - \log(q)
\]

### Summary of the Simplified Expressions:

1. \(\log_6(5/x) = \log_6(5) - \log_6(x)\)
2. \(\log_2(z/32) = \log_2(z) - 5\)
3. \(\ln(x^3/m) = 3 \ln(x) - \ln(m)\)
4. \(\ln(x/4a) = \ln(x) - 2 \ln(2) - \ln(a)\)
5. \(\log(a^4/pq) = 4 \log(a) - \log(p) - \log(q)\)

Let me know if you need anything else!