Moderate -0.5 This is a straightforward application of logarithm laws (subtraction rule and power rule) followed by basic algebraic manipulation to make y the subject. It requires fewer steps than a typical multi-part question and involves only routine techniques with no problem-solving insight needed.
Use law for the logarithm of a quotient or a product
M1
Remove logarithms and obtain \(yx^2 = y + 5\), or equivalent
A1
Obtain answer \(y = \frac{5}{x^2 - 1}\)
A1
Total: [4]
| State or imply $2 \ln x = \ln(x^2)$ | B1 |
| Use law for the logarithm of a quotient or a product | M1 |
| Remove logarithms and obtain $yx^2 = y + 5$, or equivalent | A1 |
| Obtain answer $y = \frac{5}{x^2 - 1}$ | A1 |
**Total: [4]**
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2 It is given that $\ln ( y + 5 ) - \ln y = 2 \ln x$. Express $y$ in terms of $x$, in a form not involving logarithms.
\hfill \mbox{\textit{CAIE P2 2009 Q2 [4]}}