Exponential Functions and Logarithms Flashcard Review

Front

What is the formula for calculating the effective annual rate (EAR) from the nominal interest rate compounded quarterly?

Back

The formula is: \( EAR = \left(1 + \frac{r}{n}\right)^{nt} - 1 \) where \( r \) is the nominal rate, \( n \) is the number of compounding periods per year, and \( t \) is the number of years.

Front

What is the future value formula for compound interest?

Back

The future value (FV) is calculated using the formula: \( FV = P \left(1 + \frac{r}{n}\right)^{nt} \) where \( P \) is the principal amount, \( r \) is the annual interest rate, \( n \) is the number of times interest is compounded per year, and \( t \) is the number of years.

Front

What is the present value formula for continuous compounding?

Back

The present value (PV) for continuous compounding is given by: \( PV = FV \cdot e^{-rt} \) where \( FV \) is the future value, \( r \) is the annual interest rate, and \( t \) is the time in years.

Front

What does it mean for a function to be exponential?

Back

An exponential function is a mathematical function of the form \( f(x) = a \cdot b^x \) where \( a \) is a constant, \( b \) is the base (a positive real number), and \( x \) is the exponent.

Front

How do you determine the base of an exponential function from two points?

Back

To find the base \( b \) of an exponential function given two points \( (x_1, y_1) \) and \( (x_2, y_2) \), use the formula: \( b = \left(\frac{y_2}{y_1}\right)^{\frac{1}{x_2 - x_1}} \).

Front

What is the relationship between logarithms and exponents?

Back

Logarithms are the inverse operations of exponentiation. If \( b^y = x \), then \( \log_b(x) = y \).

Front

What is the change of base formula for logarithms?

Back

The change of base formula states: \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \) for any positive base \( k \).

Front

What is the natural logarithm?

Back

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \) (approximately 2.71828).

Front

How do you solve for \( x \) in the equation \( 2^x = 16 \)?

Back

To solve for \( x \), rewrite 16 as a power of 2: \( 2^x = 2^4 \). Therefore, \( x = 4 \).

Front

What is the value of \( f(-2) \) for the function \( f(x) = 4^{x+2} - 5 \)?

Back

To find \( f(-2) \), substitute -2 into the function: \( f(-2) = 4^{0} - 5 = 1 - 5 = -4 \).