Log_b(xy) = log_b(x) + log_b(y)

Definition

This equation represents the logarithmic property known as the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the factors. It highlights a fundamental relationship in logarithmic functions, emphasizing how multiplication within the argument translates into addition outside of the logarithm. This rule is crucial for simplifying complex logarithmic expressions and solving exponential equations efficiently.

5 Must Know Facts For Your Next Test

1. The product rule for logarithms is useful when dealing with multiplication in equations, allowing simplification into addition, which is easier to handle.
2. This property holds true for all positive bases b, where b ≠ 1, ensuring that it can be applied across different contexts involving logarithms.
3. When using this property, it is essential to remember that both x and y must be positive, as logarithms of non-positive numbers are undefined.
4. This rule can also be extended to more than two factors; for example, log_b(xyz) = log_b(x) + log_b(y) + log_b(z).
5. Understanding this property aids in solving exponential equations by converting them into linear forms, making it easier to isolate variables.

Review Questions

• How can you apply the product rule for logarithms to simplify expressions involving products of variables?
  • To apply the product rule for logarithms, if you have an expression like log_b(xy), you can rewrite it as log_b(x) + log_b(y). This helps in simplifying complex calculations involving multiplication by breaking them down into manageable parts. For example, if you're tasked with finding log_2(8 * 4), using the product rule allows you to convert it into log_2(8) + log_2(4), which you can evaluate separately.
• What restrictions exist when using the product rule for logarithms, particularly concerning the arguments of the logarithm?
  • When using the product rule for logarithms, both x and y in the expression log_b(xy) must be positive real numbers. Logarithms are undefined for zero and negative numbers, so if you attempt to use this property with such values, you will encounter mathematical errors. This restriction is crucial to ensure that all operations remain valid and meaningful within the realm of real numbers.
• Evaluate the expression log_5(25 * 5^3) using the product rule and explain each step in your solution.
  • To evaluate log_5(25 * 5^3), first apply the product rule: log_5(25 * 5^3) = log_5(25) + log_5(5^3). Next, calculate each part: since 25 = 5^2, then log_5(25) = 2. For log_5(5^3), by the power rule of logarithms, this equals 3. Now combine both results: 2 + 3 = 5. Therefore, log_5(25 * 5^3) equals 5, demonstrating how to systematically use properties of logarithms to reach a solution.

Related terms

Logarithm: A logarithm is the inverse operation to exponentiation, representing the power to which a base must be raised to obtain a certain number.

Exponential Function: An exponential function is a mathematical function in which an independent variable appears in the exponent, typically written as f(x) = b^x, where b is a constant base.

Change of Base Formula: The change of base formula allows for the conversion of logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k.