Worksheet

1

If $f(x)$ is a one-to-one function, what is the value of $f^{-1}(f(x))$?

$x$ $0$ $1$ $f(x)^2$

2

Find the inverse function, $f^{-1}(x)$, for the function $f(x) = 3x - 5$.

$3x+5$ $\frac{x}{3} + 5$ $5x+3$ $\frac{x+5}{3}$

3

If the point $(4, 9)$ lies on the graph of a one-to-one function $g(x)$, which point must lie on the graph of its inverse, $g^{-1}(x)$?

$(4, 9)$ $(9, 4)$ $(\frac{1}{4}, \frac{1}{9})$ $(-4, -9)$

4

What must be true for a function $f(x)$ to have an inverse function $f^{-1}(x)$?

It must be an even function. It must be an odd function. It must be a continuous function. It must be one-to-one (injective).

5

Find the inverse function of $f(x) = 2x - 5$.

$f^{-1}(x) = \frac{x + 5}{2}$ $f^{-1}(x) = 5x - 2$ $f^{-1}(x) = 2x + 5$ $f^{-1}(x) = \frac{x - 5}{2}$

6

The graphs of a function $f(x)$ and its inverse $f^{-1}(x)$ are symmetric with respect to which line?

The line $y = -x$ The line $y = x$ The y-axis The x-axis

7

If the domain of a one-to-one function $f(x)$ is $[1, 5]$ and its range is $[10, 20]$, what is the range of $f^{-1}(x)$?

$(1, 5)$ $(10, 20)$ $[1, 5]$ $[10, 20]$

8

If $f(x)$ and $f^{-1}(x)$ are inverse functions, which of the following expressions is always equal to $x$?

$f(f(x))$ $f(x) \cdot f^{-1}(x)$ $f(f^{-1}(x))$ $f^{-1}(f^{-1}(x))$

9

The general form of an exponential function is $y = b^x$. What restriction is placed on the base $b$ for the function to be classified as exponential?

$b > 0$ and $b \neq 1$. $b$ is any real number. $b$ must be an integer. $b \neq 0$.

10

Which of the following equations represents a valid exponential function?

$y = (\frac{1}{x})^3$ $y = 0.8^x$ $y = x^5$ $y = (-3)^x$

11

For the basic exponential function $f(x) = b^x$ (where $b > 0$ and $b \neq 1$), what is the value of the function when $x=0$?

$b$ $0$ $1$ $-1$

12

Why is $y = 1^x$ generally excluded from the definition of an exponential function?

It simplifies to the constant function $y=1$. It is undefined when $x$ is negative. It always passes through the origin $(0, 0)$. It is not defined for irrational values of $x$.

13

Evaluate the exponential function $f(x) = 5^x$ when $x = 3$.

$15$ $625$ $125$ $25$

14

Which exponential equation is equivalent to the logarithmic equation $\log_4 64 = 3$?

$64^3 = 4$ $4^3 = 64$ $3^4 = 64$ $\sqrt[3]{64} = 4$

15

Write the exponential equation $10^5 = 100,000$ in logarithmic form.

$\log_{10} 5 = 100,000$ $\log_{100,000} 5 = 10$ $\log_{5} 10 = 100,000$ $\log_{10} 100,000 = 5$

16

Evaluate $\log_2 32$.

16 4 6 5

17

What is the value of $x$ if $\log_{7} x = 2$?

\frac{2}{7} 49 9 14

18

If $\log_b 125 = 3$, what is the base $b$?

5 4 2 10

19

Evaluate $\log_8 1$.

\frac{1}{8} 0 8 1

20

What is the equivalent logarithmic form of $a^c = b$?

$\log_b a = c$ $\log_c a = b$ $\log_c b = a$ $\log_a b = c$

21

Evaluate $\log_{10} 10000$.

4 3 10 5

22

Evaluate $\log_3 \left(\frac{1}{9}\right)$.

3 -2 2 \frac{1}{2}

23

What is the value of $y$ in the equation $5^y = 625$?

4 5 2 3

24

Which exponential statement is equivalent to $\log_6 6 = 1$?

$1^6 = 6$ $6^6 = 1$ $1^1 = 6$ $6^1 = 6$

25

Find the value of $\log_9 3$.

\frac{1}{2} -1 2 3

26

The expression $\log_{1/2} 8 = -3$ is equivalent to which exponential form?

$8^{-3} = \frac{1}{2}$ $8^{1/2} = -3$ $(\frac{1}{2})^{-3} = 8$ $-3^{1/2} = 8$

27

If $2^x = 128$, which logarithmic equation correctly represents this relationship?

$\log_2 x = 128$ $\log_2 128 = x$ $\log_{128} 2 = x$ $\log_x 2 = 128$

28

Find the value of $\log_{100} 10$.

2 10 -2 \frac{1}{2}

29

Which exponential equation is equivalent to the logarithmic equation $\log_{\frac{1}{4}} 16 = -2$?

$\left( \frac{1}{4} \right)^{-2} = 16$ $16^{-2} = \frac{1}{4}$ $\left( -2 \right)^{16} = \frac{1}{4}$ $\left( -2 \right)^{\frac{1}{4}} = 16$

30

The exponential expression $81^{\frac{1}{4}} = 3$ can be rewritten in logarithmic form as:

$\log_{3} 81 = \frac{1}{4}$ $\log_{81} 3 = \frac{1}{4}$ $\log_{\frac{1}{4}} 3 = 81$ $\log_{81} 4 = 3$

31

Which logarithmic expression is equivalent to $3^{2k+1} = M$?

$\log_3 M = 2k+1$ $\log_{M} 3 = 2k+1$ $\log_{2k+1} M = 3$ $\log_3 (2k+1) = M$

32

Determine the value of $y$ in the expression $\log_5 \left( \frac{1}{125} \right) = y$.

$y = \frac{1}{3}$ $y = 25$ $y = -3$ $y = 3$

33

If $\log_b 64 = -3$, what is the base $b$?

$b = \frac{1}{4}$ $b = 4$ $b = 2$ $b = -4$

34

Which logarithmic form correctly represents the equation $e^{x^2} = 5$?

$\log_e 5 = x^2$ $\log_5 (x^2) = e$ $\log_e x = 5^2$ $\log_{x^2} 5 = e$

35

Convert the equation $\log_{\sqrt{2}} (x+1) = 4$ into its equivalent exponential form.

$4^{\sqrt{2}} = x+1$ $(\sqrt{2})^{(x+1)} = 4$ $(\sqrt{2})^4 = x+1$ $(x+1)^4 = \sqrt{2}$

36

Which logarithmic equation is equivalent to $\left( \frac{1}{9} \right)^x = 27$?

$\log_{27} x = \frac{1}{9}$ $\log_x \left( \frac{1}{9} \right) = 27$ $\log_x 27 = \frac{1}{9}$ $\log_{\frac{1}{9}} 27 = x$

37

If $\log_{0.5} 8 = y$, what is the value of $y$?

$y = 3$ $y = -3$ $y = -4$ $y = 4$

38

The logarithmic equation $\log_A B = \frac{C}{D}$ is equivalent to which exponential form?

$B^A = \frac{C}{D}$ $\left( \frac{C}{D} \right)^A = B$ $B^{\frac{C}{D}} = A$ $A^{\frac{C}{D}} = B$

39

Convert the logarithmic equation $\log_{3}(x) = \frac{2}{5}$ to its equivalent exponential form.

$3^{\frac{2}{5}} = x$ $3^{x} = \frac{2}{5}$ $x^{\frac{2}{5}} = 3$ $(\frac{2}{5})^{3} = x$

40

Which logarithmic equation is equivalent to $y^{-3} = 64$?

$\log_{-3}(y) = 64$ $\log_{y}(-3) = 64$ $\log_{y}(64) = -3$ $\log_{64}(y) = -3$

41

Convert the natural logarithmic equation $\ln(2x - 1) = 5$ into exponential form.

$e^{5} = 2x - 1$ $10^{5} = 2x - 1$ $5^{e} = 2x - 1$ $e^{(2x-1)} = 5$

42

Determine the correct logarithmic form for the equation $(x+1)^3 = 2y - 5$.

$\log_{3}(x+1) = 2y - 5$ $\log_{3}(2y - 5) = x+1$ $\log_{2y-5}(3) = x+1$ $\log_{(x+1)}(2y - 5) = 3$

43

If $\log_{(2a+1)}(100) = b-1$, what is the corresponding exponential equation?

$(b-1)^{100} = 2a+1$ $(2a+1)^{b-1} = 100$ $(2a+1)^{100} = b-1$ $100^{b-1} = 2a+1$

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CTJan27 Online Year 9 - Introduction to Logarithms