Standard form

Standard form

Standard form, also known as scientific notation, is a way of expressing numbers as a product of two parts:

• The digits (with the decimal point placed after the first digit), followed by
• × 10 raised to a power, which shows how many places to move the decimal point.

How to write a number in standard form?

To figure out the power of 10, think of how many decimal places to move:

• If the number is 10 or greater, the decimal point moves to the left, and the power of 10 will be positive. For example: $$47055 = 4.7055 \times 10^4$$
• If the number is smaller than 1, the decimal point moves to the right, and the power of 10 will be negative. For example: $$0.00025 = 2.5 \times 10^{-4}$$
• If the number is already between 1 and 10, the power of 10 is 0. For example: $$4.5 = 4.5 \times 10^0$$

Note: After converting a number into scientific notation, ensure that the digits part is between 1 and 10 (it can be 1 but never 10). The power of 10 shows exactly how many places to move the decimal point.

Computation involving multiplication and division of numbers in standard form

When performing computations with numbers in standard form (scientific notation), follow these rules:

• Multiplication of numbers in standard form: Multiply the digits and add the exponents. $$(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$$
• Division of numbers in standard form: Divide the digits and subtract the exponents. $$\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$$

Logarithms

A logarithm answers the question: How many of one number do we multiply to get another number? For example, how many 2s do we multiply to get 16? Solution: We multiply four 2s to get 16. So, the logarithm of 16 with base 2 is 4. In notation, we can write this as:

$$\log_2 16 = 4$$

Base, exponent, and logarithm

In the logarithmic expression $\log_b y = x$, the components are:

• Base: The number we are multiplying (in this case, 2).
• Exponent: The power to which the base is raised (in this case, 4).
• Result: The number we want to get (in this case, 16).

The general relationship between exponents and logarithms is:

$$a^x = y \quad \text{in logarithmic form is:} \quad \log_a y = x$$

Example 1: Write the following statements in logarithmic form

Example:

$$2^3 = 8$$

In logarithmic form, this would be:

$$\log_2 8 = 3$$

Example 2: Write the following statements in logarithmic form

$$10^4 = 10000$$

The logarithmic form is:

$$\log_{10} 10000 = 4$$

Example 3: Write the following statements in logarithmic form

$$5^2 = 25$$

The logarithmic form is:

$$\log_5 25 = 2$$