Powers and Logarithms in Mathematics

Powers of 2

• $2^{-1} = \frac{1}{2}$, $3^{-1} = \frac{1}{3}$, $5^{-1} = \frac{1}{5}$, $7^{-1} = \frac{1}{7}$, $9^{-1} = \frac{1}{9}$, $10^{-1} = \frac{1}{10}$
• $2^{\frac{1}{2}} = \sqrt{2}$
• $2^{\frac{2}{3}} = \sqrt[3]{4}$
• $2^2 = 4$, $2^3 = 8$, $2^4 = 16$

Cubes of Numbers

• $1^3 = 1$, $2^3 = 8$, $3^3 = 27$, $4^3 = 64$, $5^3 = 125$, $6^3 = 216$, $7^3 = 343$, $8^3 = 512$, $9^3 = 729$, $10^3 = 1000$

Table of Powers of 2

• The table lists values of $2^n$ from $2^0$ to $2^{10}$.
• $2^0 = 1$, $2^1 = 2$, $2^2 = 4$, $2^3 = 8$, $2^4 = 16$, $2^5 = 32$, $2^6 = 64$, $2^7 = 128$, $2^8 = 256$, $2^9 = 512$, $2^{10} = 1024$

Table of Powers

• The table shows values for $n^1$ to $n^{10}$ for various values of n (3, 4, 5, 6, 7, 8, 9, 10)
• The table is incomplete and contains missing data

Logarithm Fundamentals

• Logarithm tables and calculators can be used to determine logarithm values
• Logarithms are defined with different bases, including log₁₀, log₂, log₃.

Practice Problems

• The document includes examples and worked problems to help practice calculating logarithms
• The problems involve applying log rules to simplify expressions, such as log(a * b) = log a + log b, log(a / b) = log a - log b, log(ax) = x * log a, loga(x) = log x / log a, log₁₀ x = log x.
• Example calculations include:
  • log(625) = log(5⁴) = 4 * log 5
  • log(1029) = log(10² * 10⁹ * 10⁻⁴³) = 2 + 9 + (-4)3
  • log(5 x 10⁻¹) = log 5 + log 10⁻¹ = 0.7 - 1
  • log(6 x 10⁷) = log 6 + log 10⁷ = 0.78 + 7
  • log(6 x 10⁻¹) = log 6 + log 10⁻¹ = 0.78 - 1
• Some of the calculations in the examples are incomplete and marked with ....

This quiz covers the fundamentals of powers, specifically exploring the powers of 2 and cubes of numbers. It also includes logarithmic concepts and their applications. Test your understanding of these essential mathematical topics.