Logarithmic Functions Quiz

Questions and Answers

Which of the following is the correct domain for the logarithmic function $f(x) = ext{log}(x-3)$?

$x < 3$

$x > 3$ (correct)

All real numbers

$x \geq 3$

What is the range of the function $g(x) = ext{log}(x)$?

Negative real numbers

Non-negative real numbers

Positive real numbers

All real numbers (correct)

When condensing the expression $3 ext{log}(x) + 2 ext{log}(y)$, which of the following is the correct form?

log($3xy^2$)

log($x^3 y^2$) (correct)

log($x^2y^3$)

log($xy^3)$

What is the horizontal asymptote of the logarithmic function $h(x) = ext{log}(x)$?

<p>y = 0</p> Signup and view all the answers

Which equation correctly represents solving the logarithmic equation $\text{log}_2(x) = 3$?

<p>$x = 2^3$</p> Signup and view all the answers

Study Notes

Asymptotes, Domain, and Range of Logarithmic Functions

Logarithmic functions have vertical asymptotes where the argument of the logarithm equals zero.
The general form of a logarithmic function is f(x) = log_b(x - h) + k, where (h, k) translates the graph.
The domain of a logarithmic function is x > h; it includes all x-values greater than the horizontal shift.
The range of a logarithmic function is all real numbers, (-∞, ∞), indicating it can take any vertical value.

Expanding and Condensing Logarithmic Functions

To expand logarithmic expressions, use properties such as:
- log_b(MN) = log_b(M) + log_b(N)
- log_b(M/N) = log_b(M) - log_b(N)
- log_b(M^p) = p * log_b(M)
Condensing logarithmic expressions involves combining logs using the above properties.
Example of condensing: log_b(M) + log_b(N) = log_b(MN)

Solving Logarithmic Equations

To solve logarithmic equations, if possible, rewrite the equation in exponential form: if log_b(x) = y, then b^y = x.
Check for extraneous solutions, especially when manipulating logarithmic expressions.
Common strategies include isolating the logarithm, using properties of logarithms, and applying one-to-one properties of logarithmic functions.

Description

Test your understanding of logarithmic functions in this quiz. You will be asked to find asymptotes, determine the range and domain, and expand and condense logarithmic expressions. Additionally, you'll solve various logarithmic equations to demonstrate your skills.