Exponential functions with other bases

Writing a^x as e^(x ln a), differentiating and integrating a^x, and why e is the natural base for calculus.

Changing to base $e$

Since $a = e^{\ln a}$, any exponential can be written with base $e$: $$a^x = \big(e^{\ln a}\big)^x = e^{x\ln a}.$$

Example 1●

Write $2^x$ as a power of $e$.

$2^x = e^{x\ln 2} \approx e^{0.693x}$

Other bases

$$\frac{d}{dx}a^x = a^x\ln a \qquad \int a^x\,dx = \frac{a^x}{\ln a} + C$$

Example 2●

Find the gradient of $y = 3^x$ at $x = 2$.

$\dfrac{dy}{dx} = 3^x\ln 3$, so at $x = 2$ the gradient is $9\ln 3$.

Example 3●●

Evaluate $\displaystyle\int_0^2 2^x\,dx$.

$\Big[\dfrac{2^x}{\ln 2}\Big]_0^2 = \dfrac{4 - 1}{\ln 2} = \dfrac{3}{\ln 2}$

Example 4●●

Differentiate $y = 5^{2x}$.

$5^{2x} = e^{2x\ln 5}$, so $\dfrac{dy}{dx} = 2\ln 5 \times 5^{2x}$.

Flashcards

$a^x$ in base $e$

$e^{x\ln a}$

$dfrac{d}{dx}a^x$

$a^x\ln a$

$displaystyleint a^x,dx$

$\dfrac{a^x}{\ln a} + C$

Practice