Questions and Answers
What is the result of log(base b)b^k?
k
What is the logarithmic form of b^y = x?
log(base b)x = y
What is the evaluation step for log(base 32)1/4?
32^x = 1/4; (2^5)^x = 1/2^2; 2^5x = 2^-2; 5x = -2
What is the natural logarithm definition?
Signup and view all the answers
What is the common logarithm definition?
Signup and view all the answers
What is the exponential form of log(base b)x = y?
Signup and view all the answers
What is the result of lne^x?
Signup and view all the answers
What is the result of log10^x?
Signup and view all the answers
What is the result of b^log(base b)x?
Signup and view all the answers
What is the result of log1?
Signup and view all the answers
What is the compound interest formula?
Signup and view all the answers
What is the continuous compound interest formula?
Signup and view all the answers
What is the product property of logarithms?
Signup and view all the answers
What is the quotient property of logarithms?
Signup and view all the answers
What is the power property of logarithms?
Signup and view all the answers
What is the value of log(base 25)5?
Signup and view all the answers
What is the value of log(base 7)1/7?
Signup and view all the answers
What is the value of log(base 2)8?
Signup and view all the answers
How would you evaluate y = lnx + 7?
Signup and view all the answers
What does 'n' represent in finance?
Signup and view all the answers
How do you undo logs in an equation?
Signup and view all the answers
How do you undo 'e' in an equation?
Signup and view all the answers
How do you undo 'ln' in an equation?
Signup and view all the answers
What are the guidelines for solving exponential equations?
Signup and view all the answers
What are the guidelines for solving logarithmic equations?
Signup and view all the answers
Study Notes
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).
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This set of flashcards covers key concepts related to logarithms in Algebra 2. You'll learn definitions, conversions between logarithm and exponential forms, and methods for evaluating logarithmic expressions. Perfect for students looking to strengthen their understanding of logarithmic functions.
