Leaving Cert Mathematics Logs and Indices: Given $\log_a 2 = p$ and $\log

Given $\log_a 2 = p$ and $\log_a 3 = q$, where $a > 0$, write each of the following in terms of $p$ and $q$: (i) $\frac{8}{\log_a 3}$ - Leaving Cert Mathematics - Question b - 2016

Question b

$Given-$\log_a-2-=-p$-and-$\log_a-3-=-q$,-where-$a->-0$,-write-each-of-the-following-in-terms-of-$p$-and-$q$:--(i)-$\frac{8}{\log_a-3}$-Leaving Cert Mathematics-Question b-2016.png$

Given $\log_a 2 = p$ and $\log_a 3 = q$, where $a > 0$, write each of the following in terms of $p$ and $q$: (i) $\frac{8}{\log_a 3}$. (ii) $\log_a \frac{9a^2}{16}... show full transcript

Worked Solution & Example Answer:Given $\log_a 2 = p$ and $\log_a 3 = q$, where $a > 0$, write each of the following in terms of $p$ and $q$: (i) $\frac{8}{\log_a 3}$ - Leaving Cert Mathematics - Question b - 2016

Step 1

(i) $\frac{8}{\log_a 3}$

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Answer

To express $\frac{8}{\log_a 3}$ in terms of $p$ and $q$ , we can start with the known logarithm.

Since $\log_a 3 = q$ , we can rewrite the expression as follows:

$\frac{8}{\log_a 3} = \frac{8}{q}$

Next, we know that $\log_a 8$ can be expressed in terms of $\log_a 2$ :

$\log_a 8 = \log_a (2^3) = 3\log_a 2 = 3p$

Therefore, we can write:

$\frac{8}{\log_a 3} = 3p - q$

Step 2

(ii) $\log_a \frac{9a^2}{16}$

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Answer

Using the properties of logarithms:

First, we split the logarithm:

$\log_a \frac{9a^2}{16} = \log_a(9a^2) - \log_a(16)$

Next, we express each term separately:

For $\log_a(9a^2)$ $lo g_{a} (9 a^{2})$ :
- This can be split as: $\log_a 9 + \log_a a^2$
- From $\log_a 9$ , since $9 = 3^2$ , we have: $\log_a 9 = \log_a(3^2) = 2\log_a 3 = 2q$
- And since $\log_a a^2 = 2$ (that is, the logarithm of a number to its own base): $\log_a a^2 = 2$

Combining these gives us:

$\log_a(9a^2) = 2q + 2$

For $\log_a(16)$ $lo g_{a} (16)$ :
- Since $16 = 2^4$ , we have: $\log_a 16 = \log_a(2^4) = 4\log_a 2 = 4p$

Combining the two parts:

$\log_a \frac{9a^2}{16} = (2q + 2) - 4p = 2q + 2 - 4p$

Given $\log_a 2 = p$ and $\log_a 3 = q$, where $a > 0$, write each of the following in terms of $p$ and $q$: (i) $\frac{8}{\log_a 3}$ - Leaving Cert Mathematics - Question b - 2016

Worked Solution & Example Answer:Given $\log_a 2 = p$ and $\log_a 3 = q$, where $a > 0$, write each of the following in terms of $p$ and $q$: (i) $\frac{8}{\log_a 3}$ - Leaving Cert Mathematics - Question b - 2016

(i) $\frac{8}{\log_a 3}$

(ii) $\log_a \frac{9a^2}{16}$

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