Algebra 2 Logs Flashcards

Logarithmic Definitions and Properties

• Logarithm Basics: log(base b)b^k = k; this illustrates that the logarithm of a number to its own base equals the exponent.
• Conversion to Logarithmic Form: The equation b^y = x can be expressed as log(base b)x = y, highlighting the relationship between exponential and logarithmic forms.
• Logarithmic Evaluation: Evaluating log(base 32)1/4 involves transforming into exponential form, leading to the equation 32^x = 1/4.

Types of Logarithms

• Natural Logarithm: Represented as log(base e)x = lnx, applicable for x > 0; ln denotes the natural logarithm.
• Common Logarithm: Defined as log(base 10)x = logx, also for x > 0; common logarithms use base 10.

Exponential and Logarithmic Transformations

• Exponential Forms: From log(base b)x = y, it can be rewritten as b^y = x, clarifying the conversion between logarithmic and exponential expressions.
• Natural Logarithm Evaluation: lne^x = x; the natural log of e raised to x yields x.
• Common Logarithm Evaluation: log10^x = x; similar to natural logs, the common log of 10 raised to x simplifies to x.

Logarithmic Identities

• Identity with Base: The expression b^log(base b)x = x indicates that raising the base to its logarithm returns the original value.
• Logarithm of 1: log1 = 0; any base raised to the power of 0 equals 1.

Financial Formulas Involving Exponentials

• Compound Interest Formula: Defined as A = P(1 + r/n)^nt, where A is total amount, P is principal, r is interest rate, n is number of times interest applied per year, and t is time in years.
• Continuous Compound Interest: Expressed as A = Pe^rt; it models interest accrued continuously over time.

Properties of Logarithms

• Product Property: log(base b)uv = log(base b)u + log(base b)v; allows decomposition of products into sums of logs.
• Quotient Property: log(base b)u/v = log(base b)u - log(base b)v; allows transformation of division into subtraction of logs.
• Power Property: log(base b)u^n = n * log(base b)u; simplifies calculations when dealing with powers in logarithmic expressions.

Specific Logarithmic Evaluations

• Logarithmic Values:
  • log(base 25)5 = 1/2 shows the relationship of different bases.
  • log(base 7)1/7 = -1 demonstrates results for inverses.
  • log(base 2)8 = 3 indicates the logarithm where base 2 raised to 3 equals 8.

Solving Equations with Logs and Exponents

• Solving Exponential Equations:
  • Isolate the exponential term, apply a logarithm to both sides, and solve for the variable.
• Logarithmic Equation Guidelines:
  • Condense the logarithmic expression when possible, isolate it, apply the base of the log to both sides, and solve for the variable.

Undoing Logarithmic Operations

• Undoing Logs: To reverse a logarithm, apply the base to the result (e.g., log's base).
• Undoing Natural Log: To reverse ln, use e raised to the exponent.
• Undoing Exponential Functions: For expressions involving e, apply ln to revert.

Compounding Frequency

• n: Represents the number of times interest is compounded within a year (e.g., annually, quarterly, monthly, daily).