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

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Calculus with Other Bases
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Change-of-base formula for log⁡ax\log_a x

log⁡ax=log⁡exlog⁡ea\log_a x = \frac{\log_e x}{\log_e a}

Derivative of log⁡ax\log_a x

1xlog⁡ea\frac{1}{x \log_e a}

Express aa as power of ee

a=elog⁡eaa = e^{\log_e a}

Express axa^x in base ee

ax=exlog⁡eaa^x = e^{x \log_e a}

Derivative of axa^x

axlog⁡eaa^x \log_e a

Integral of axa^x

axlog⁡ea+C\frac{a^x}{\log_e a} + C

Gradient at x-intercept of y=log⁡exy = \log_e x

11

Gradient at x-intercept of y=log⁡axy = \log_a x

1log⁡ea\frac{1}{\log_e a}

When diff./int. axa^x, multiply/divide by what constant?

log⁡ea\log_e a

Restriction on base aa

Positive and not equal to 11

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