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### Problem 7:
**Selesaikanlah.**

---

**a.**  
\[
\text{Jika } \frac{\log a}{2} = m \text{ dan } \frac{\log a}{3} = n, \text{ tentukanlah hubungan antara m dan n.}
\]
This problem is asking for the relationship between \( m \) and \( n \), given that:

\[
m = \frac{\log a}{2} \quad \text{and} \quad n = \frac{\log a}{3}
\]

To solve this, we can express \( \log a \) in terms of \( m \) and \( n \):

\[
m = \frac{\log a}{2} \implies \log a = 2m
\]
\[
n = \frac{\log a}{3} \implies \log a = 3n
\]

Now, since both expressions represent \( \log a \), we equate them:

\[
2m = 3n
\]

Thus, the relationship between \( m \) and \( n \) is:

\[
m = \frac{3}{2}n
\]

---

**b.**  
\[
\text{Diketahui } \log 2^5 = p, \text{ nyatakanlah } \log 2 \text{ dalam } p.
\]
Given that \( \log 2^5 = p \), we can simplify the logarithm using logarithmic rules:

\[
\log 2^5 = 5 \log 2
\]

So:

\[
5 \log 2 = p
\]

To find \( \log 2 \) in terms of \( p \):

\[
\log 2 = \frac{p}{5}
\]

---

**c.**  
\[
\text{Jika } x > 0 \text{ dan } x = 1, \log 6 = a \text{ dan } \log 8 = b, \text{ tentukan nilai dari } 3a - b.
\]
We are asked to find \( 3a - b \), given that \( \log 6 = a \) and \( \log 8 = b \).

Using logarithmic properties, we can express \( a \) and \( b \):

\[
\log 6 = \log (2 \cdot 3) = \log 2 + \log 3
\]
\[
\log 8 = \log 2^3 = 3 \log 2
\]

Thus:

\[
a = \log 2 + \log 3
\]
\[
b = 3 \log 2
\]

Now, substitute these into \( 3a - b \):

\[
3a - b = 3(\log 2 + \log 3) - 3 \log 2
\]

Simplify:

\[
3a - b = 3 \log 2 + 3 \log 3 - 3 \log 2 = 3 \log 3
\]

So:

\[
3a - b = 3 \log 3
\]

---

**d.**  
\[
\text{Jika } \log 5 = p, \text{ tentukan nilai } \log_{10} 125.
\]
To find \( \log_{10} 125 \), note that:

\[
125 = 5^3
\]

Thus:

\[
\log 125 = \log 5^3 = 3 \log 5
\]

Given that \( \log 5 = p \), we substitute:

\[
\log 125 = 3p
\]

---

**e.**  
\[
\text{Jika } \log \left(\frac{b}{a}\right)^4 = x, \text{ tentukan nilai dari } \log \left(\frac{a^2}{b}\right).
\]
We are given that \( \log \left(\frac{b}{a}\right)^4 = x \). Using logarithmic rules:

\[
\log \left(\frac{b}{a}\right)^4 = 4 \log \left(\frac{b}{a}\right)
\]

Now, we need to find \( \log \left(\frac{a^2}{b}\right) \). Using logarithmic properties:

\[
\log \left(\frac{a^2}{b}\right) = \log a^2 - \log b = 2 \log a - \log b
\]

We can express \( \log \left(\frac{b}{a}\right) \) as:

\[
\log \left(\frac{b}{a}\right) = \log b - \log a
\]

Now, solve for \( \log \left(\frac{a^2}{b}\right) \) in terms of \( x \).

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Do you want more details on any specific step or concept?

Here are 5 related questions:
1. Can you explain more about logarithmic properties, such as \( \log a^b = b \log a \)?
2. How do we simplify logarithms involving products and quotients?
3. What are the steps to change the base of a logarithm?
4. How do we solve equations involving logarithmic expressions?
5. Can you apply these logarithmic rules to natural logarithms?

**Tip**: Always check if logarithmic terms can be simplified using product, quotient, or power rules.

Solve the following logarithmic problems:

Explore the solution to logarithmic equations involving log properties such as the power rule and quotient rule. Learn how to relate logarithmic terms and simplify expressions step by step. Suitable for Grades 10-12.

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