5.4
Laws of Logarithms I
Logarithms are fundamental mathematical operations that serve as the inverse of exponentiation. They provide a means to express how many times a base…
A logarithm indicates how many times a base must be multiplied by itself to reach a specific number. Base 10 logarithms are usually written as 'log'.
Two important types of logarithmic laws are the product law and the quotient law.
The product law states that the logarithm of a product equals the sum of the logarithms.
For example, the logarithm of 10 times 10 is the logarithm of 100, which equals 2. According to the product rule, this equals the sum of the logarithms of 10 and 10 — both of which are 1, because the logarithm of a number to its own base is 1, giving a total of 2.
The quotient law states that the logarithm of a quotient equals the difference of the logarithms.
For example, the logarithm of 10 divided by 10 is the logarithm of 1, which equals 0 since the logarithm of 1 to any base is 0. The quotient rule shows this equals the difference between the logarithms of 10 and 10 — giving 0.
The pH scale, for instance, uses logarithms to measure acidity from hydrogen ion concentration. If the concentration is 2 times 10 to the power of negative 7 mol/L, pH is the negative logarithm of that. The product law simplifies this to 6.7.
View the full transcript and gain access to JoVE Core videos
Q1: What does a logarithm tell you about a number?
A logarithm indicates how many times a base must be multiplied by itself to reach a specific number. For base 10, written as log, the logarithm expresses the exponent needed. For example, log(100) = 2 because 10 must be multiplied by itself twice to equal 100. This inverse relationship to exponentiation makes logarithms essential for simplifying complex calculations.
Q2: How does the product law of logarithms work?
The product law states that the logarithm of a product equals the sum of the logarithms. For example, log(10 × 10) = log(10) + log(10) = 1 + 1 = 2, which equals log(100). This rule breaks down large values into manageable components, streamlining computations without direct multiplication and simplifying complex expressions.
Q3: What is the quotient law of logarithms?
The quotient law states that the logarithm of a quotient equals the difference of the logarithms. For instance, log(10 ÷ 10) = log(10) − log(10) = 1 − 1 = 0, which equals log(1). This property aligns with the fact that the logarithm of 1 to any base is always zero, since any number raised to the power of zero equals one.
Q4: Why is the logarithm of 1 always zero?
The logarithm of 1 to any base is always zero because any number raised to the power of zero equals one. This fundamental property is verified through the quotient law: log(1) = log(b ÷ b) = log(b) − log(b) = 0. This relationship holds regardless of the base used in the logarithmic expression.
Q5: How are logarithms used to measure pH?
The pH scale uses logarithms to measure acidity from hydrogen ion concentration. pH is calculated as the negative logarithm of the hydrogen ion concentration. For example, with a concentration of 2 × 10⁻⁷ mol/L, the product law simplifies the calculation to yield a pH of 6.7, demonstrating how logarithmic laws streamline real-world scientific measurements.
Q6: What real-world applications use logarithmic scales?
Logarithmic scales appear in acoustics and seismology, where sound levels are measured in decibels and earthquake magnitudes on the Richter scale. Both systems employ logarithmic scales because each unit increase represents a substantial rise in intensity or energy, reflecting the nonlinear behavior of these phenomena and making large ranges manageable.
Q7: Why must logarithmic arguments always be positive?
Logarithms are undefined for non-positive values, so the arguments of all logarithmic expressions must remain positive. This domain condition is critical when solving equations with logarithms, as it determines which solutions are mathematically valid. Understanding this restriction is essential when using exponential equations with logarithms problem solving.