Understanding Logarithms and Cube Roots: A Guide to Solving log 3 √3
In mathematics, log 3 √3 represents a logarithmic expression involving the cube root of 3. Understanding logarithms and cube roots is essential to solve this problem and many others in algebra and calculus.
What are Logarithms?
A logarithm answers the question: "To what power must we raise the base to get a certain number?". For example, the logarithm of 100 to the base 10 is 2, because 10 raised to the power of 2 equals 100. This can be written mathematically as:
log₁₀ 100 = 2
What are Cube Roots?
A cube root is a number that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2, because 2 x 2 x 2 = 8. This can be written mathematically as:
∛8 = 2
Solving log 3 √3
To solve log 3 √3, we need to determine the power to which we must raise 3 to get the cube root of 3. Let's break it down step by step:
- Understanding √3: The cube root of 3 (√3) represents a number that, when multiplied by itself three times, equals 3.
- Expressing √3 in Exponential Form: We can express the cube root of 3 in exponential form as 3¹/³.
- Logarithmic Equation: Therefore, our equation becomes log₃ (3¹/³) = ?.
- Solving for the Exponent: We need to find the power (x) that we must raise 3 to get 3¹/³: 3ˣ = 3¹/³.
- Solution: Since the bases are the same, the exponents must be equal. Therefore, x = 1/3.
Therefore, log 3 √3 = 1/3.
Key Points to Remember:
- The logarithm of a number to a specific base is the exponent to which the base must be raised to obtain that number.
- The cube root of a number is a number that, when multiplied by itself three times, equals the original number.
- When solving logarithmic expressions, remember the properties of logarithms and exponents.
Example:
Consider another example: log₂ 8.
- We know that 2³ = 8.
- Therefore, log₂ 8 = 3.
Conclusion
Understanding logarithms and cube roots is crucial for solving mathematical problems involving these concepts. In the case of log 3 √3, by expressing the cube root in exponential form and applying the properties of logarithms, we found that the solution is 1/3.