Algebra Class: Logarithms and Exponentials

The exponent to which a base must be raised to obtain a specific number. For example, log₂ 8 = 3 because 2³ = 8.

Logarithms and exponentials are inverse operations. This means that if you apply one operation and then the other, you get back to the original value. For example, 2³ = 8 and log₂ 8 = 3. Applying the logarithm after the exponent gives you back the original exponent.

Using algebraic manipulation, logarithms and exponentials can be used to solve equations involving each other. For example, to solve 2ˣ = 8, we can take the logarithm base-2 of both sides to get x = log₂ 8 = 3.

The graphs of logarithms and exponentials are symmetrical about the line y = x. This symmetry demonstrates the inverse relationship.

Converting a logarithmic equation to exponential form involves rearranging the equation to isolate the base raised to the exponent. Example: log₂ 8 = 3 becomes 2³ = 8.

Converting an exponential equation to logarithmic form involves rearranging the equation to express the exponent as a logarithm. Example: 2³ = 8 becomes log₂ 8 = 3.

Evaluating logarithms without a calculator involves understanding the relationship between logarithms and exponents. For example, log₂ 16 = 4 because 2⁴ = 16.

Using properties of logarithms like product rule, quotient rule, and power rule, we can rewrite an expression as a sum, difference, or multiple of logarithms. Example: log₂ (8x) can be rewritten as log₂ 8 + log₂ x.

Using properties of logarithms, we can condense multiple logarithms into a single logarithm. Example: log₂ 8 + log₂ x can be rewritten as log₂ (8x).

Solving exponential equations algebraically involves isolating the variable using logarithm operations. Example: 2ˣ = 16. Applying log₂ to both sides gives x = log₂ 16 = 4.