Unit 8-9 Exponential and Logarithmic Functions content and lessons

What is the general form of an exponential function?

y = b^x, where b > 0.

When does an exponential function exhibit growth?

When the base (b) is greater than 1.

What happens to exponential functions with bases between 0 and 1?

They model exponential decay.

What is the domain of an exponential function?

All real numbers, x ∈ R.

What is the range of an exponential function like y = 2^x?

y > 0.

How does the graph of y = 2^x behave?

It rises steeply as x increases.

What is the y-intercept of any exponential function?

(0, 1).

Describe the horizontal asymptote of an exponential function.

y = 0; the function approaches but never reaches this line.

How does the value of the function y = 2^x change as x decreases?

It approaches 0, but does not become negative.

How do we evaluate the function y = log_a(x)?

It is the power to which base 'a' must be raised to obtain x.

What is the relationship between an exponential function and its logarithmic function?

They are inverses of each other.

What is the product rule of logarithms?

log_a(m * n) = log_a(m) + log_a(n).

What is the quotient rule of logarithms?

log_a(m / n) = log_a(m) - log_a(n).

What is the power rule of logarithms?

log_a(m^k) = k * log_a(m).

Describe what happens to the values of 'y' in the function y = (1/2)^x as x increases.

The value of y decreases towards 0.

What does the notation log_a(b) represent?

It represents the exponent to which base 'a' must be raised to yield 'b'.

How do you convert the equation b^y = x to logarithmic form?

y = log_b(x).

What is the domain of the logarithmic function y = log_a(x)?

x > 0.

What happens to logarithmic functions as their arguments approach zero?

They approach negative infinity.

State the change of base formula for logarithms.

log_a(b) = log_c(b) / log_c(a).

What is an asymptote in terms of logarithmic functions?

The line x = 0 (the y-axis) is a vertical asymptote.

Identify the range of y = log_a(x).

All real numbers, y ∈ R.

What is the relationship between x and y in the context of solving x^y = b?

You can isolate y using logarithms: y = log_b(x).

Explain the concept of exponential decay.

Exponential decay occurs when a quantity decreases at a rate proportional to its value.

Provide an example of an equation expressing exponential decay.

N(t) = N_0 * (1 - r)^t.

What does the variable 'r' represent in exponential growth and decay formulas?

The growth or decay rate, expressed as a decimal.

How do you find the time it takes for a population to double using exponential growth formulas?

Use the formula: t = (log(2)) / (log(1 + r)).

What is the initial amount in the context of exponential functions?

The value of the quantity before growth or decay begins.

Describe an example where logarithmic functions are applied in real life.

Logarithmic functions are used to measure the pH of solutions or the intensity of earthquakes.

What is the y-intercept of the exponential decay function y = a(1 - r)^x, assuming r > 0?

(0, a), where a is the initial amount.

How can you graphically interpret exponential growth?

By plotting points for increasing y values corresponding with increasing x, illustrating steep growth.

What is the purpose of logarithms in scientific calculations?

They simplify the process of dividing and multiplying large numbers by converting them into addition and subtraction.

How do logarithmic transformations assist in solving exponential equations?

They allow for isolation of the variable x, transforming exponential relationships into linear forms.

What kind of problems commonly use exponential functions?

In problems involving populations, finance, and radioactive decay.