Exponential and Logarithms Review

Front

What is the general form of an exponential function?

Back

An exponential function can be expressed as f(t) = a(1 + r)^t, where 'a' is the initial value, 'r' is the growth rate, and 't' is time.

Front

How do you find the initial population in an exponential growth model?

Back

The initial population is the value of P when t = 0. For P = 6191(1.04)^t, the initial population in 1990 (t=0) is 6191.

Front

What does the base of an exponential function represent?

Back

The base of an exponential function indicates the growth factor. For example, in P = 6191(1.04)^t, the base 1.04 shows a 4% increase each year.

Front

What is the relationship between exponential functions and logarithms?

Back

Logarithms are the inverse operations of exponential functions. If y = a^x, then log_a(y) = x.

Front

How do you solve an exponential equation using logarithms?

Back

To solve an exponential equation, take the logarithm of both sides. For example, to solve e^(x+6) + 5 = 7, isolate the exponential term and apply the natural logarithm.

Front

What is the natural logarithm?

Back

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.718.

Front

How do you convert an exponential equation to logarithmic form?

Back

To convert a^b = c to logarithmic form, write log_a(c) = b.

Front

What is the value of log_8(4x + 4) = 2 in terms of x?

Back

To solve log_8(4x + 4) = 2, convert to exponential form: 4x + 4 = 8^2, which simplifies to 4x + 4 = 64, leading to x = 15.

Front

What is the formula for continuous growth?

Back

The formula for continuous growth is P(t) = Pe^(rt), where P is the initial amount, r is the growth rate, and t is time.

Front

What does a growth rate of 4% mean in an exponential model?

Back

A growth rate of 4% means that the quantity increases by 4% of its current value each time period.