Let's work through the questions from the image you've provided.

Problem 7:

Selesaikanlah.

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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:

$m = \frac{3}{2}n$

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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 $p$:

$\log 2 = \frac{p}{5}$

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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:

$3a - b = 3 \log 3$

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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:

$\log 125 = 3p$

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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.