Integer Exponents and Logarithm

Study Notes

Integer exponents involve using whole numbers as exponents, which can be positive or negative, to indicate how many times a base number is multiplied by itself. Logarithms answer the question of how many times one number must be multiplied together to make another number, with the base indicating the number being multiplied.

• Integer Exponents — Exponents that are integers, either positive or negative. Example: In $2^3$, the exponent is 3, meaning 2 is multiplied by itself 3 times: $2 \times 2 \times 2 = 8$.
• Fractional Exponents — Exponents that are fractions, indicating roots. Example: $9^{1/2}$ is the square root of 9, which is 3.
• Logarithms — The power to which a base number must be raised to produce a given number. Example: $\log_2(8) = 3$ because $2^3 = 8$.
• Common Logarithms — Logarithms with base 10, often written without a base. Example: $\log(100) = 2$ because $10^2 = 100$.

Exam Tips

Key Definitions to Remember

• Integer exponents are whole numbers used as exponents.
• Logarithms determine how many times a base is multiplied to reach a number.
• Common logarithms have a base of 10.

Common Confusions

• Confusing the base of a logarithm with the number being calculated.
• Mixing up positive and negative exponents.

Typical Exam Questions

• What is $\log_2(64)$? Answer: 6, because $2^6 = 64$.
• Express $5^3 = 125$ in logarithm form. Answer: $\log_5(125) = 3$.
• Find $x$ if $\log_5(x-7) = 1$. Answer: $x = 12$.

What Examiners Usually Test

• Understanding of converting between exponential and logarithmic forms.
• Ability to evaluate expressions with integer and fractional exponents.
• Solving equations involving logarithms.