# Natural Exponential

We learnt that the simplest exponential functions are $y=a^x$ where $a>0$, $a \ne 1$.For all positive values of the base $a$, the graph is always positive, that is $a^x > 0$ for all $a>0$.There are an infinite number of possible choices for the base number. However, where exponential data is examined in engineering, science, and […]

# Exponential Equations (Indicial Equations)

The equation $a^x=y$ is an example of a general exponent equation (indicial equation), and $2^x = 32$ is an example of a more specific exponential equation (indicial equation). To solve one of these equations, it is necessary to write both sides of the equation with the same base if the unknown is an exponent (index) […]

# Division using Exponents (Indices)

If we are given $a^8 \div a^3$, we can also write this as $\dfrac{a^8}{a^3}$, which means $\dfrac{a \times a \times a \times a \times a \times a \times a \times a}{a \times a \times a}$.As there are $8$ factors of $a$ on the top line (numerator), and $3$ factors of $a$ on the bottom line […]

# Multiplication using Exponents (Indices)

If we wish to calculate $5^4 \times 5^3$, we could write in factor form to get:\begin{align} \displaystyle5^4 \times 5^3 &= (5 \times 5 \times 5 \times 5) \times (5 \times 5 \times 5) \\&= 5^7 \end{align} Example 1 Simplify $7^2 \times 7^3$ after first writing in factor form. \( \begin{align} \displaystyle7^2 \times 7^3 […]