Exponential and Logarithmic Functions Quiz

Which condition indicates that an exponential function is exhibiting growth?

What is the equation of the exponential function represented by the points (0, 4) and (1, 20)?

Which of the following logarithmic expressions is invalid?

What is the result of condensing the expression -ln(x) + 3ln(y) - 6ln(z)?

What is the y-intercept of the function g(x) = log₅(x - 2) + 1?

What is the inverse of the function h(x) = 5log₃(x + 2)?

If $b$ in an exponential equation is such that $0 < b < 1$, what type of behavior does the function exhibit?

Which logarithmic identity can be applied to expand the expression log(27x⁵y⁻¹z)?

For the equation $y = 4(5)^x$, what does the point (1, 20) signify?

What is the range of the function g(x) = log₅(x - 2) + 1?

Exponential Growth/Decay Equations

To determine if an equation represents exponential growth or decay, examine the base (b).
If b > 1, it's exponential growth.
If 0 < b < 1, it's exponential decay.

Exponential Function Equations from Points

Given two points (x₁, y₁) and (x₂, y₂), find the exponential equation.
First, solve for b using the formula b = (y₂/y₁)^(1/(x₂-x₁)).
Then, substitute b back into either point to find 'a' (the initial value).

Evaluating Logarithmic Expressions

log₅1 = 0
log₂2 = 1
log₁₀ 10 = 1

Expanding Logarithms

log(27x⁵y⁴z) = log27 + logx⁵ + logy⁴ + logz This expands to log27 + 5logx + 4logy + logz

Condensing Logarithms

log(x + 1) - log(x) + log(2) = log((x + 1)(2)/x) or log((2x + 2)/x)
-ln(x) + 3ln(y) - 6ln(z) = ln( y³ / (xz⁶))
-log₃x - log₃8 - 2log₃y² = log₃(1/(8x y⁴))

Inverse Functions

To find the inverse of a function, swap x and y, and solve for y. Example: To find the inverse of g(x) = 3x + 4, swap x and y giving x = 3y + 4. Then solve for y to get y = (x-4)/3.

Logarithmic Function Graphs

Graphs of logarithmic functions have a vertical asymptote, which is a vertical line the graph approaches but never touches. This will be x=2 (specific to log₅(x – 2) + 1).
The range of a logarithmic function is typically all real numbers (-∞, ∞).
The domain of a logarithmic function is usually restricted to positive values of the argument (x − 2 > 0). Note specific cases show alternative domain restrictions.
Examples shown are of functions that are increasing over their defined domains only.

Test your knowledge on exponential growth and decay equations, as well as logarithmic expressions and their properties. This quiz covers key concepts such as evaluating, expanding, and condensing logarithms, along with finding inverse functions. Challenge yourself and see how well you understand these crucial mathematical topics!