Calculus with Other Bases (HSC SSCE Mathematics Advanced): Quizzes

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Calculus with Other Bases
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What is logax\log_a x expressed in terms of logex\log_e x using the change-of-base formula?

logexlogea\frac{\log_e x}{\log_e a}

What constant is logax\log_a x multiplied by to equal logex\log_e x?

logea\log_e a

What is ddxlogax\frac{d}{dx}\log_a x?

1xlogea\frac{1}{x \log_e a}

Which logarithmic function has gradient exactly 1 at its x-intercept?

y=logexy = \log_e x only

What is the gradient of y=logaxy = \log_a x at the x-intercept (1,0)?

1logea\frac{1}{\log_e a}

How do you express axa^x in terms of base ee?

exlogeae^{x \log_e a}

What is ddxax\frac{d}{dx}a^x?

axlogeaa^x \log_e a

What is axdx\int a^x \, dx?

axlogea+C\frac{a^x}{\log_e a} + C

Why are calculus formulas simplest when a=ea = e?

Because logee=1\log_e e = 1

When working with axa^x, you multiply by logea\log_e a for what operation?

Differentiation

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