6.1 The functions a^x and e^x — Practise with help

1. $\dfrac{x^8}{x^3} = x^{8-3} = x^5$.
2. $27^{1/3} = 3$, so $27^{2/3} = 3^2 = 9$.
3. $(2x)^3 = 2^3x^3 = 8x^3$.

If question 2 was hard, revise fractional indices in Chapter 2 before you continue.

The base $5$ is larger than $1$, so the curve rises. Start from the point $(0, 1)$.

Plot $(0, 1)$, $(1, 5)$ and $(-1, 0.2)$, then join them with a smooth rising curve that approaches the $x$-axis on the left.

The curve rises because $5 > 1$. It passes through $(0, 1)$. The asymptote is $y = 0$, because $5^x$ approaches zero as $x$ becomes very negative but stays positive. The range is $y > 0$.

A negative index turns the fraction over. Use $\left(\dfrac{1}{4}\right)^{-2} = 4^2$.

Work out each value separately: flip the fraction for the negative index, and remember that any positive number to the power $0$ is $1$.

The values fall as $x$ rises, because the base is between $0$ and $1$.

Write $\left(\dfrac{1}{2}\right)^x$ as $2^{-x}$, and compare the two exponents.

Replacing $x$ with $-x$ in $y = 2^x$ gives $y = 2^{-x}$. Recall what that transformation does to a graph.

This is the reflection of $y = 2^x$ in the $y$-axis. Both curves pass through $(0, 1)$ and both have asymptote $y = 0$. The first rises and the second falls.

Test the claim at $x = 0$, at $x = 1$, and at a fractional value such as $x = \dfrac{1}{2}$.

Look up the definition of $x^{1/2}$, and decide whether $0^{1/2}$ can be a fixed positive number.

The value $0^0$ is not defined, and $0^x = 0$ for every positive $x$. So the claim fails at $x = 0$ and gives $0$ rather than $1$ for every other $x$. A fractional power such as $0^{1/2}$ is also $0$, so the base gives no growth and no decay. This is why the base of an exponential function must be positive and cannot be $1$.

First write the base as a power of $e$: $7 = e^{\ln 7}$. Then raise both sides to the power $x$.

Use the index law $\left(e^{m}\right)^x = e^{mx}$ to move the $x$ into the exponent.

So $k = \ln 7 = 1.94591\ldots$, which is $1.95$ to three significant figures.

The value is $e \times e \times e$, which is a little more than $2.7^3$.

Multiply $e$ by itself three times, or use the power button on your calculator, and then round to three significant figures.

which is $20.1$ to three significant figures.