Q: Which of the following best describes the graph of $f(x) = \log_a x$ for $0 < a < 1$?

The graph is decreasing for $x > 0$ and has a vertical asymptote at $x = 0$.

Question 1

Which of the following statements correctly describes the relationship between the exponential function $f(x) = a^x$ and its inverse?

Accepted Answer

The inverse of $f(x) = a^x$ is $f^{-1}(x) = \log_a x$, and $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$.

Question 2

Convert the logarithmic equation $\log_5 (2x - 1) = 3$ to its equivalent exponential form.

Accepted Answer

$2x - 1 = 5^3$

Question 3

Which of the following is the correct domain of the function $f(x) = \log_a x$ where $a > 0$ and $a 
eq 1$?

Accepted Answer

$(0, +\infty)$

Question 4

Evaluate $\log_2 \left( \dfrac{1}{16} \right)$ without a calculator.

Accepted Answer

$-4$

Question 5

Which property of logarithms allows you to write $\log_a (xy)$ as a sum?

Accepted Answer

$\log_a (xy) = \log_a x + \log_a y$

Question 6

Solve for $x$: $e^{2x} = 7$.

Accepted Answer

$x = \dfrac{1}{2} \ln 7$

Question 7

Which of the following is the correct change of base formula for logarithms?

Accepted Answer

$\log_a x = \dfrac{\log_b x}{\log_b a}$

Question 8

If $\log_3 x + \log_3 (x+1) = \log_3 12$, what is the solution for $x$?

Accepted Answer

$x = 3$

Question 9

Which of the following best describes the range of the function $f(x) = \log_a x$?

Accepted Answer

$(-\infty, +\infty)$

Question 10

Rewrite the expression $2 \log_5 x + \log_5 y - 3 \log_5 z$ as a single logarithm.

Accepted Answer

$\log_5 \left( \dfrac{x^2 y}{z^3} \right)$

Question 11

A certain radioactive substance decays according to the formula $A(t) = A_0 e^{-kt}$. If after 5 years, only 60% of the original amount remains, what is the value of $k$? (Round to 3 decimal places.)

Accepted Answer

$k = 0.102$

Question 12

Which of the following is NOT a property of logarithms?

Accepted Answer

$\log_a (x + y) = \log_a x + \log_a y$

Question 13

Solve for $x$: $4^x = 32$.

Accepted Answer

$x = \dfrac{5}{2}$

Question 14

If $\ln x = 2$, what is the value of $x$?

Accepted Answer

$x = e^2$

Question 15

Which of the following points is on the graph of $f(x) = \log_2 x$?

Accepted Answer

$(8, 3)$

Question 16

Given $\log_{10} 142 = 2.1523$ and $\log_{10} 10 = 1$, what is $\log_{10} (1420)$?

Accepted Answer

$3.1523$

Question 17

Which of the following best describes the graph of $f(x) = \log_a x$ for $0 < a < 1$?

Accepted Answer

The graph is decreasing for $x > 0$ and has a vertical asymptote at $x = 0$.

Question 18

If $\log_a x = y$, which of the following is true?

Accepted Answer

$a^y = x$

Question 19

Solve for $x$: $\ln (x^2) = 4$.

Accepted Answer

$x = e^2$ or $x = -e^2$

Logarithmic Functions and Their Properties in Business Calculus

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