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

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Calculus with Other Bases
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What is log⁡ax\log_a x expressed in terms of log⁡ex\log_e x using the change-of-base formula?

log⁡exlog⁡ea\frac{\log_e x}{\log_e a}

What constant is log⁡ax\log_a x multiplied by to equal log⁡ex\log_e x?

log⁡ea\log_e a

What is ddxlog⁡ax\frac{d}{dx}\log_a x?

1xlog⁡ea\frac{1}{x \log_e a}

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

y=log⁡exy = \log_e x only

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

1log⁡ea\frac{1}{\log_e a}

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

exlog⁡eae^{x \log_e a}

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

axlog⁡eaa^x \log_e a

What is ∫ax dx\int a^x \, dx?

axlog⁡ea+C\frac{a^x}{\log_e a} + C

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

Because log⁡ee=1\log_e e = 1

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

Differentiation

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