9709 · 2.2
Logarithmic and exponential functions flashcards
Revision flashcards for Cambridge 9709 Logarithmic and exponential functions (syllabus 2.2). Flip, recall, then mark a real past-paper question.
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What is the relationship between $y = a^x$ and $x = \log_a y$?
They are equivalent statements. The logarithm, $\log_a y$, gives you the exponent, $x$, that the base $a$ must be raised to in order to get $y$.
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What is the base of the natural logarithm, $\ln x$?
The base is the mathematical constant $e$, where $e \approx 2.71828$. So, $\ln x$ is shorthand for $\log_e x$.
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What is the value of $\log_a a$?
1, because $a^1 = a$. The log of the base itself is always 1.
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What is the value of $\log_a 1$?
0, because $a^0 = 1$. The log of 1 is always 0 for any valid base.
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State the product rule for logarithms.
$\log_a(xy) = \log_a x + \log_a y$. The log of a product is the sum of the logs.
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State the quotient rule for logarithms.
$\log_a(x/y) = \log_a x - \log_a y$. The log of a quotient is the difference of the logs.
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State the power rule for logarithms.
$\log_a(x^n) = n \log_a x$. The exponent inside the log can be brought out as a multiplier.
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What is the change of base formula for logarithms?
$\log_a b = \frac{\log_c b}{\log_c a}$. This is useful for calculators which often only have base 10 (log) or base e (ln).
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Common mistake: What is $\log(x+y)$?
It is NOT $\log x + \log y$. There is no simplification rule for the logarithm of a sum or difference.
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How do you solve an equation like $5^x = 100$?
Take logarithms of both sides: $x \log 5 = \log 100$, so $x = \frac{\log 100}{\log 5}$. You can use any base, but `ln` or `log` on a calculator is easiest.
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What does a negative value for $k$ in $y = Ae^{kt}$ signify?
It signifies exponential decay, where the quantity decreases over time, approaching a limit. A positive $k$ signifies exponential growth.
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What is the domain of $y = \ln(x)$?
$x > 0$. You cannot take the logarithm of zero or a negative number.