Question 1

Solve the equation. $\left( 3 ^ { x + 5 } \right) \left( 3 ^ { x } \right) = 81$
A) $x = \frac { 3 } { 2 }$
B) $x = \frac { 9 } { 2 }$
C) $x = - \frac { 1 } { 2 }$
D) $x = - \frac { 7 } { 2 }$

Accepted Answer

C

Question 2

An equation of the form $f ( x ) = b ^ { x }$ is a one-to-one function.

Accepted Answer

The function $f(x) = b ^ x$ is one-to-one because for any two distinct inputs, say $x_1$ and $x_2$, we have $b ^ {x_1} \neq b ^ {x_2}$, and therefore the outputs $b ^ {x_1}$ and $b ^ {x_2}$ are also distinct. This means that there are no two inputs that produce the same output, which is the definition of a one-to-one function.

Question 3

Solve the equation. $10 ^ { x } = 0.1$
A) x = 3
B) x = $-$ 3
C) x = 1
D) x = 2
E) x = $-$ 1

Accepted Answer

We can rewrite 0.1 as a fraction:

$$ 0.1 = \frac{1}{10} $$

Then we can rewrite the equation as:

$$ 10^x = \frac{1}{10} $$

Taking the logarithm of both sides with base 10:

$$ log_{10}(10^x) = log_{10}\left(\frac{1}{10}\right) $$

$$ x = \log_{10}\left(\frac{1}{10}\right) = \log_{10}(1) - \log_{10}(10) = 0 - 1 = -1 $$

Therefore, the solution is x = -1, which corresponds to answer choice E.

Question 4

Solve the equation. $10 ^ { x } = 0.000001$
A) x = 8
B) x = $-$ 6
C) x = 6
D) x = 7
E) x = $-$ 8

Accepted Answer

The equation $10^x = 0.000001$ can be rewritten as $10^x = 10^{-6}$. Comparing the exponents of both sides gives $x = -6$.

Question 5

The graphs of $f ( x ) = 4 ^ { x }$ and $f ( x ) = \left( \frac { 1 } { 4 } \right) ^ { x }$ are reflections of each other across the x axis.

Accepted Answer

The graphs of $f(x) = 4^x$ and $f(x) = \left(\frac{1}{4}\right)^x$ are reflections of each other across the y-axis, not the x-axis. This is because they are inverse functions in terms of their base, but their exponents remain the same, affecting their growth and decay in opposite directions along the y-axis.

Question 6

If $b > 0$ and $b \neq 1$ , then the function f defined by $f ( x ) = b ^ { x }$ where x is any real number, is called the power function with base b .

Accepted Answer

The function $f(x) = b^x$ where $b > 0$ and $b \neq 1$ is called the exponential function with base $b$, not the power function. Power functions have the form $f(x) = x^a$ where $a$ is any real number.

Question 7

Solve the equation. $\left( \frac { 1 } { 6 } \right) ^ { x } = \frac { 1 } { 36 }$
A) x = $-$ 2
B) x = 2
C) x = $-$ 4
D) x = 4
E) x = $-$ 3

Accepted Answer

The answer of Solve the equation. \(\left( \frac { 1...

Question 8

All graphs of $f ( x ) = b ^ { - x }$ contain the point $( 1,0 )$ .

Accepted Answer

If we substitute $x=1$ into the function, we get $f(1)=b^{-1}=\frac{1}{b}$, which is never equal to zero for any value of $b$. Therefore, the point $(1,0)$ cannot be on the graph of $f(x)=b^{-x}$. This statement is false.

Question 9

Solve the equation. $81 ^ { x } = 243$
A) $x = - \frac { 5 } { 4 }$
B) $x = \frac { 9 } { 4 }$
C) $x = \frac { 5 } { 4 }$
D) $x = \frac { 1 } { 4 }$

Accepted Answer

We can rewrite 81 as $3^4$, and 243 as $3^5$. Therefore, the equation becomes:

$$3^{4x} = 3^5$$

Dividing both sides by $3^4$, we get:

$$3^{4x-4} = 3$$

Since $3^1=3$, we have:

$$4x-4=1$$

Solving for $x$, we get:

$$4x = 5 \Rightarrow x = \frac{5}{4}$$

Therefore, the answer is choice C.

Question 10

Solve the equation. $\left( 256 ^ { x } \right) \left( 4 ^ { 12 x - 1 } \right) = 64$
A) $x = \frac { 1 } { 4 }$
B) $x = \frac { 9 } { 4 }$
C) $x = \frac { 5 } { 4 }$
D) $x = - \frac { 5 } { 4 }$

Accepted Answer

First, express all numbers as powers of 2: $256 = 2^8$, $4 = 2^2$, and $64 = 2^6$. The equation becomes $(2^{8x})(2^{2(12x-1)}) = 2^6$. Simplify the exponents: $2^{8x + 24x - 2} = 2^6$, leading to $32x - 2 = 6$. Solving for $x$, we get $x = \frac{1}{4}$.

Question 11

Solve the equation. $\left( 3 ^ { 4 x - 4 } \right) \left( 3 ^ { x + 1 } \right) = 27$
A) $x = - \frac { 6 } { 5 }$
B) $x = \frac { 11 } { 5 }$
C) $x = - \frac { 1 } { 5 }$
D) $x = \frac { 6 } { 5 }$

Accepted Answer

\begin{align*}
\left( 3 ^ { 4 x - 4 } ight) \left( 3 ^ { x + 1 } ight) &= 27 \
3^{4x-4+x+1}&=27 \
3^{5x-3}&= 27 \
3^{5x-3}&= 3^ {3} \
5x-3&= 3 \
x&=\frac{6}{5}
\end{align*}

Thus, the answer is choice D, as it is the only option that includes this value of $x$.

Question 12

Graph the exponential function. $f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }$
A) 
B) 
C)

Accepted Answer

The answer of Graph the exponential function. \(f ( x...

Question 13

Solve the equation. $4 ^ { 2 x - 4 } = \frac { 1 } { 256 }$
A) x = 2
B) x = 0
C) x = 1
D) x = $-$ 2

Accepted Answer

The answer of Solve the equation. \(4 ^ { 2...

Question 14

Solve the equation. $4 ^ { 2 x } = 16$
A) x = 1
B) x = $-$ 1
C) x = $-$ 3
D) x = 3
E) x = $-$ 2

Accepted Answer

We can rewrite 16 as $4 ^{2}$. Thus, the equation becomes:

$$4 ^ { 2 x } = 4 ^{2}$$
Simplifying, we get:

$$2x = 2$$
Dividing by 2 on both sides, we get:

$$x = 1$$

Therefore, the answer is choice A.

Question 15

Solve the equation. $3 ^ { x } = 27$
A) x = $-$ 3
B) x = 4
C) x = 3
D) x = $-$ 5
E) x = 5

Accepted Answer

We know that 3 raised to what power gives 27, which is 3. Hence, x should be equal to 3.

Question 16

Graph the exponential function. $f ( x ) = 4 ^ { x }$
A) 
B) 
C)

Accepted Answer

The answer of Graph the exponential function. \(f ( x...

Question 17

Solve the equation. $2 ^ { - x } = \frac { 1 } { 2 }$
A) x = 2
B) x = 4
C) x = $-$ 1
D) x = $-$ 3
E) x = 1

Accepted Answer

Taking the base-2 logarithm of both sides, we get $-x = -1$, which gives us $x=1$. Therefore, the answer is E.

Question 18

Solve the equation. $27 ^ { 38 x } = 9 ^ { x + 7 }$
A) $x = - \frac { 9 } { 8 }$
B) $x = \frac { 1 } { 8 }$
C) $x = \frac { 17 } { 8 }$
D) $x = \frac { 9 } { 8 }$

Accepted Answer

We can rewrite the equation in terms of a common base:
$$27^{38x} = 9^{x+7} \Rightarrow (3^3)^{38x} = (3^2)^{x+7} \Rightarrow 3^{114x} = 3^{2x+14} $$
Now we can set the exponents equal to each other:
$$114x = 2x+14 \Rightarrow 112x = 14 \Rightarrow x = \frac{1}{8}$$
Therefore, the answer is choice B.

Question 19

Solve the equation. $( 256 ) \left( 4 ^ { x } \right) = 64 ^ { x }$
A) x = $-$ 2
B) x = 4
C) x = $-$ 4
D) x = 3
E) x = 2

Accepted Answer

Begin by noticing that $256=4^4$ and $64=4^3$. Thus the given equation can be rewritten as 
$$(4^4)\cdot(4^x)=(4^3)^x$$
Simplifying the expression on the right gives $4^{3x}$. Thus we now have
$$4^{4+x}=4^{3x}$$
We can now solve this equation for $x$ by equating the exponents:
$$4+x=3x$$
$$2x=4$$
$$x=2$$
Therefore, the correct answer is E.

Question 20

Solve the equation. $4 ^ { 2 x - 3 } = 64$
A) x = 3
B) x = $-$ 3
C) x = $-$ 5
D) x = 5
E) x = 4

Accepted Answer

We can write the equation as:

$4^{2x-3}=4^3$

By comparing the exponents, we get:

$2x-3=3$

Solving for $x$, we get:

$2x=6$

$x=3$

Therefore, the solution is $x=3$, which corresponds to choice $	extbf{(A)}$.

Quiz 14: Exponential and Logarithmic Functions

Solve the equation. (3x+5) (3x) =81\left( 3 ^ { x + 5 } \right) \left( 3 ^ { x } \right) = 81(3x+5) (3x) =81

An equation of the form f(x)=bxf ( x ) = b ^ { x }f(x)=bx is a one-to-one function.

Solve the equation. 10x=0.110 ^ { x } = 0.110x=0.1

Solve the equation. 10x=0.00000110 ^ { x } = 0.00000110x=0.000001

The graphs of f(x)=4xf ( x ) = 4 ^ { x }f(x)=4x and f(x)=(14)xf ( x ) = \left( \frac { 1 } { 4 } \right) ^ { x }f(x)=(41​)x are reflections of each other across the x axis.

If b>0b > 0b>0 and b≠1b \neq 1b=1 , then the function f defined by f(x)=bxf ( x ) = b ^ { x }f(x)=bx where x is any real number, is called the power function with base b .

Solve the equation. (16) x=136\left( \frac { 1 } { 6 } \right) ^ { x } = \frac { 1 } { 36 }(61​) x=361​

All graphs of f(x)=b−xf ( x ) = b ^ { - x }f(x)=b−x contain the point (1,0)( 1,0 )(1,0) .

Solve the equation. 81x=24381 ^ { x } = 24381x=243

Solve the equation. (256x) (412x−1) =64\left( 256 ^ { x } \right) \left( 4 ^ { 12 x - 1 } \right) = 64(256x) (412x−1) =64

Solve the equation. (34x−4) (3x+1) =27\left( 3 ^ { 4 x - 4 } \right) \left( 3 ^ { x + 1 } \right) = 27(34x−4) (3x+1) =27

Graph the exponential function. f(x) =(13) xf ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }f(x) =(31​) x

Solve the equation. 42x−4=12564 ^ { 2 x - 4 } = \frac { 1 } { 256 }42x−4=2561​

Solve the equation. 42x=164 ^ { 2 x } = 1642x=16

Solve the equation. 3x=273 ^ { x } = 273x=27

Graph the exponential function. f(x) =4xf ( x ) = 4 ^ { x }f(x) =4x

Solve the equation. 2−x=122 ^ { - x } = \frac { 1 } { 2 }2−x=21​

Solve the equation. 2738x=9x+727 ^ { 38 x } = 9 ^ { x + 7 }2738x=9x+7

Solve the equation. (256) (4x) =64x( 256 ) \left( 4 ^ { x } \right) = 64 ^ { x }(256) (4x) =64x

Solve the equation. 42x−3=644 ^ { 2 x - 3 } = 6442x−3=64

Solve the equation. $\left( 3 ^ { x + 5 } \right) \left( 3 ^ { x } \right) = 81$

An equation of the form $f ( x ) = b ^ { x }$ is a one-to-one function.

Solve the equation. $10 ^ { x } = 0.1$

Solve the equation. $10 ^ { x } = 0.000001$

The graphs of $f ( x ) = 4 ^ { x }$ and $f ( x ) = \left( \frac { 1 } { 4 } \right) ^ { x }$ are reflections of each other across the x axis.

If $b > 0$ and $b \neq 1$ , then the function f defined by $f ( x ) = b ^ { x }$ where x is any real number, is called the power function with base b .

Solve the equation. $\left( \frac { 1 } { 6 } \right) ^ { x } = \frac { 1 } { 36 }$

All graphs of $f ( x ) = b ^ { - x }$ contain the point $( 1,0 )$ .

Solve the equation. $81 ^ { x } = 243$

Solve the equation. $\left( 256 ^ { x } \right) \left( 4 ^ { 12 x - 1 } \right) = 64$

Solve the equation. $\left( 3 ^ { 4 x - 4 } \right) \left( 3 ^ { x + 1 } \right) = 27$

Graph the exponential function. $f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }$

Solve the equation. $4 ^ { 2 x - 4 } = \frac { 1 } { 256 }$

Solve the equation. $4 ^ { 2 x } = 16$

Solve the equation. $3 ^ { x } = 27$

Graph the exponential function. $f ( x ) = 4 ^ { x }$

Solve the equation. $2 ^ { - x } = \frac { 1 } { 2 }$

Solve the equation. $27 ^ { 38 x } = 9 ^ { x + 7 }$

Solve the equation. $( 256 ) \left( 4 ^ { x } \right) = 64 ^ { x }$

Solve the equation. $4 ^ { 2 x - 3 } = 64$