Logarithms
Basic Principles of Logarithms
- Understand a logarithm as the inverse operation to exponentiation.
- Know that loga b = c means that a raised to the power of c equals b (a^c = b)
- Recognize that the base of the logarithm is the base of the corresponding exponential equation.
Properties of Logarithms
- Identify the product rule: loga(bc) = loga b + loga c
- Employ the quotient rule: loga(b/c) = loga b - loga c
- Utilize the power rule: loga(b^c) = c * loga b
- Remember that loga 1 = 0 for any base a
- Understand that the logarithm of a number to the same base equals 1 (loga a = 1)
- Also remember that loga a^b = b
Natural Logarithms
- Recognize ln as the natural logarithm, which is a logarithm to the base e (Euler’s number, approximately 2.71828)
- Understand that ln e = 1 and ln 1 = 0
Change of Base Formula
- For any positive number a, b, and c, where a ≠ 1 and b ≠ 1, logb a = logc a / logc b
- Use this formula to calculate logarithms with bases other than 10 or e
Solving Exponential Equations using Logarithms
- To solve exponential equations, express the equation in terms of a common base or use logarithms
- For example, to solve 2^x = 8, rewrite 8 as 2 cubed to find that x = 3.
Logarithmic Scales
- Know that logarithmic scales are used in measuring phenomena that cover a great range of values, for example, pH for acidity, the Richter scale for earthquakes, and decibels for sound intensity.