Given that $p = \log_c x$, express $\log_c (\sqrt{x} + \log_c (cx))$ in terms of $p$. - Leaving Cert Mathematics - Question b - 2012
Question b
Given that $p = \log_c x$, express $\log_c (\sqrt{x} + \log_c (cx))$ in terms of $p$.
Worked Solution & Example Answer:Given that $p = \log_c x$, express $\log_c (\sqrt{x} + \log_c (cx))$ in terms of $p$. - Leaving Cert Mathematics - Question b - 2012
Step 1
Express \( \log_c \sqrt{x} \)
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Answer
( \log_c \sqrt{x} = \log_c x^{1/2} = \frac{1}{2} \log_c x = \frac{1}{2} p )
Step 2
Express \( \log_c (cx) \)
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Answer
( \log_c (cx) = \log_c c + \log_c x = 1 + \log_c x = 1 + p )
Step 3
Combine the expressions
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Answer
( \log_c \left( \sqrt{x} + \log_c (cx) \right) = \frac{1}{2} p + (1 + p) = \frac{1}{2} p + 1 + p = \frac{3}{2} p + 1 )
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