9709 · 3.2
Logarithmic and exponential functions flashcards
Revision flashcards for Cambridge 9709 Logarithmic and exponential functions (syllabus 3.2). Flip, recall, then mark a real past-paper question.
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What is the relationship between $y=e^x$ and $y=\ln x$?
They are inverse functions. The graph of $y=\ln x$ is a reflection of the graph of $y=e^x$ in the line $y=x$.
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What is the domain and range of $y = e^x$?
Domain: $x \in \mathbb{R}$ (all real numbers). Range: $y > 0$.
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What is the domain and range of $y = \ln x$?
Domain: $x > 0$. Range: $y \in \mathbb{R}$ (all real numbers).
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State the product rule for logarithms.
$\ln(ab) = \ln a + \ln b$
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State the power rule for logarithms.
$\ln(a^n) = n \ln a$
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What is the derivative of $e^{ax}$?
$\frac{d}{dx}(e^{ax}) = ae^{ax}$
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What is the derivative of $\ln(f(x))$?
$\frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}$
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What is a common mistake when expanding $\ln(a+b)$?
A common mistake is to write $\ln a + \ln b$. This is incorrect. $\ln(a+b)$ cannot be simplified.
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How do you solve an equation of the form $e^x = c$ (where $c>0$)?
Take the natural logarithm of both sides: $\ln(e^x) = \ln c$, which simplifies to $x = \ln c$.
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What are the values of $\ln(1)$, $\ln(e)$, and $\ln(0)$?
$\ln(1) = 0$, $\ln(e) = 1$. $\ln(0)$ is undefined.
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How do you solve an equation like $a^x = b$?
Take natural logs of both sides: $\ln(a^x) = \ln b$. Then use the power rule: $x \ln a = \ln b$. Finally, solve for x: $x = \frac{\ln b}{\ln a}$.
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What does the constant $k$ represent in the model $P = P_0 e^{kt}$?
$k$ is the continuous growth rate. If $k>0$, it represents exponential growth. If $k<0$, it represents exponential decay.