Sure, let's work through the problems step-by-step.
Part (a)
We are given the equation:
[tex]\[
\log_a n = \log_a 3 + \log_a (2n - 1)
\][/tex]
To solve for [tex]\( n \)[/tex], we will use the properties of logarithms. One such property is that the sum of two logarithms with the same base can be combined into a single logarithm as follows:
[tex]\[
\log_a b + \log_a c = \log_a (b \cdot c)
\][/tex]
Applying this property to our equation, we get:
[tex]\[
\log_a n = \log_a [3 \cdot (2n - 1)]
\][/tex]
Since the logarithms on both sides have the same base, we can equate the arguments:
[tex]\[
n = 3 \cdot (2n - 1)
\][/tex]
Now, we solve this equation for [tex]\( n \)[/tex]:
[tex]\[
n = 3(2n - 1)
\][/tex]
Expanding the right-hand side:
[tex]\[
n = 6n - 3
\][/tex]
Rearranging the equation to isolate [tex]\( n \)[/tex]:
[tex]\[
n - 6n = -3
\][/tex]
[tex]\[
-5n = -3
\][/tex]
Dividing both sides by -5:
[tex]\[
n = \frac{3}{5}
\][/tex]
Thus, the value of [tex]\( n \)[/tex] is:
[tex]\[
n = 0.6
\][/tex]
Part (b)
(i) We are given [tex]\(\log_p x = 3\)[/tex]. To express [tex]\( x \)[/tex] in terms of [tex]\( p \)[/tex], we use the definition of logarithms. Recall that if [tex]\(\log_p x = y\)[/tex], then [tex]\( x = p^y \)[/tex].
Given [tex]\(\log_p x = 3\)[/tex], we can rewrite [tex]\( x \)[/tex] as:
[tex]\[
x = p^3
\][/tex]
(ii) We are also given [tex]\(\log_p y - 3 \log_p 2 = 4\)[/tex]. To simplify, we first use the property that [tex]\( k \log_p b = \log_p b^k \)[/tex]:
[tex]\[
3 \log_p 2 = \log_p (2^3) = \log_p 8
\][/tex]
So, the given equation becomes:
[tex]\[
\log_p y - \log_p 8 = 4
\][/tex]
Using the property that [tex]\(\log_p a - \log_p b = \log_p \left( \frac{a}{b} \right)\)[/tex], we can combine the logarithms:
[tex]\[
\log_p \left( \frac{y}{8} \right) = 4
\][/tex]
Using the definition of logarithms [tex]\(\log_p a = b \implies a = p^b\)[/tex], we get:
[tex]\[
\frac{y}{8} = p^4
\][/tex]
Solving for [tex]\( y \)[/tex], we multiply both sides by 8:
[tex]\[
y = 8p^4
\][/tex]
So, the expressions we found are:
- For [tex]\( x \)[/tex]:
[tex]\[
x = p^3
\][/tex]
- For [tex]\( y \)[/tex]:
[tex]\[
y = 8p^4
\][/tex]
Thus, summarizing our results:
- The value of [tex]\( n \)[/tex] is [tex]\( 0.6 \)[/tex].
- The expression for [tex]\( x \)[/tex] in terms of [tex]\( p \)[/tex] is [tex]\( p^3 \)[/tex].
- The expression for [tex]\( y \)[/tex] in terms of [tex]\( p \)[/tex] is [tex]\( 8p^4 \)[/tex].
