precal test 3

exponential decay

0 < b < 1

exponential functions can reach a maximum then decline. True or False

False. an exponential function is always increasing or decreasing.

using the definition of a logarithm to solve logarithmic equations

For any algebraic expression S and real numbers b and c, where b > 0, b ≠ 1, log_b(S) = c if and only if b^c = S

using the one-to-one property of logarithms to solve logarithmic equations

For any algebraic expressions S and T and any positive real number b, where b ≠ 1, log_b(S) = log_b(T) if and only if S = T

Does the graph of a general logarithmic function have a horizontal asymptote? Explain.

No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.

What type(s) of translation(s), if any, affect the domain of a logarithmic function?

Shifting the function right or left and reflecting the function about the y-axis will affect its domain.

The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?

Since the functions are inverses, their graphs are mirror images about the line y = x. So for every point (a, b) on the graph of a logarithmic function, there is a corresponding point (b, a) on the graph of its inverse exponential function.

exponential functions f(x)=ab^x

a CAN'T = 0 a CAN'T = 1 b > 0 b CAN'T = 1

exponential growth

b > 1

What does the change-of-base formula do? Why is it useful when using a calculator?

can be used to evaluate a logarithm with any base. For any positive real numbers M, b, and n, where n ≠ 1 and b ≠ 1.

quotient rule for logarithms

can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.

power rule for logarithms

can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.

logarithmic expression is a exponential expression in disguies

log_ba=k means b^k=a

Consider the general logarithmic function f(x) = logb (x). Why can't x be zero?

log_x is defined only x>0. log_10(0) is undefined.

Explain what log base b of 1 must be equal to and explain why.

x is the logarithm of y to the base b log_b^y is the power to which we have to raise b to get y