Front

What is exponential growth?

Back

Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to growth that accelerates over time. For example, if a population grows at a rate of 2% per year, it will increase by a larger amount each year as the base population grows.

Front

How do you calculate the future value of an investment with exponential growth?

Back

The future value can be calculated using the formula: \( FV = P(1 + r)^t \), where \( FV \) is the future value, \( P \) is the present value, \( r \) is the growth rate, and \( t \) is the time in years.

Front

What is the domain of a function?

Back

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For example, the domain of the function \( f(x) = \sqrt{x-1} \) is \( [1, \infty) \) because the square root is only defined for values of x greater than or equal to 1.

Front

What does it mean for two logarithms to be equal?

Back

If \( \log_a(b) = \log_a(c) \), then it implies that \( b = c \). This property is used to solve logarithmic equations.

Front

How do you solve the equation \( \log_4(8n-3) = \log_4(n^2+12) \)?

Back

Set the arguments equal to each other: \( 8n - 3 = n^2 + 12 \). Rearranging gives \( n^2 - 8n + 15 = 0 \), which factors to \( (n-3)(n-5) = 0 \). Thus, \( n = 3 \) or \( n = 5 \).

Front

What is the property of exponents that states \( a^m = a^n \) implies \( m = n \)?

Back

This property states that if two exponential expressions with the same base are equal, then their exponents must also be equal.

Front

How do you solve for x in the equation \( 3^3 = 3^{4x + 2} \)?

Back

Since the bases are the same, set the exponents equal: \( 3 = 4x + 2 \). Solving for x gives \( x = \frac{1}{4} \).

Front

What is the relationship between logarithms and exponents?

Back

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

Front

How do you solve the equation \( 2^x = 4^{x+1} \)?

Back

Rewrite 4 as \( 2^2 \): \( 2^x = (2^2)^{x+1} = 2^{2(x+1)} \). Set the exponents equal: \( x = 2(x + 1) \). This simplifies to \( x = -2 \).

Front

What is the formula for the growth of a population over time?

Back

The formula is \( P(t) = P_0 e^{rt} \), where \( P(t) \) is the population at time t, \( P_0 \) is the initial population, r is the growth rate, and t is time.