Standard +0.3 This is a straightforward application of logarithm laws (difference rule and power rule) followed by basic algebraic manipulation to make y the subject. It requires standard techniques with no conceptual difficulty or novel insight, making it slightly easier than average.
Use law for the logarithm of a product, quotient or power
M1
Use \(\ln e = 1\) or \(\exp(1) = 3\)
M1
Obtain correct equation free of logarithms in any form, e.g. \(\frac{y+1}{y} = ex^3\)
A1
Rearrange as \(y = (ex^3 - 1)^{-1}\), or equivalent
A1
[4 marks total]
Use law for the logarithm of a product, quotient or power | M1 |
Use $\ln e = 1$ or $\exp(1) = 3$ | M1 |
Obtain correct equation free of logarithms in any form, e.g. $\frac{y+1}{y} = ex^3$ | A1 |
Rearrange as $y = (ex^3 - 1)^{-1}$, or equivalent | A1 | [4 marks total]
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2 It is given that $\ln ( y + 1 ) - \ln y = 1 + 3 \ln x$. Express $y$ in terms of $x$, in a form not involving logarithms.
\hfill \mbox{\textit{CAIE P3 2013 Q2 [4]}}