1. change the exponential expression to an equivalent expression involving a logarithm.
$a^7 = 22$
the equivalent equation in logarithmic form is
(type an equation. use integers or fractions for any numbers in the equation.)
2. suppose that $f(x) = 7^{-x}$
(a) what is $f(4)$? when $x = 4$, what is the point on the graph of $f$?
(b) if $f(x) = \frac{1}{49}$, what is $x$? when $f(x) = \frac{1}{49}$, what is the point on the graph of $f$?
(a) $f(4) = $ the point $\quad$ is on the graph of $f$
(do not use commas in any individual coordinates.)
(b) if $f(x) = \frac{1}{49}, x = $ the point $\quad$ is on the graph of $f$
(do not use commas in any individual coordinates.)
3. simplify the expression
$\frac{(2a^9)^2}{(2a^9)^3}$
(simplify your answer.)
4. find the value(s) of the variable for which the rational expression is undefined
$\frac{5p}{p - 2}$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
$\boldsymbol{\circ}$ a. the expression is undefined for $p = $
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
$\boldsymbol{\circ}$ b. there are no values for which the expression is undefined.
5. use the negative exponent rules to simplify. write the answer with positive exponents only.
$\left(\frac{6}{5}\
ight)^{-3}$
$\left(\frac{6}{5}\
ight)^{-3} = $ (type an integer or a simplified fraction.)

Question

1. change the exponential expression to an equivalent expression involving a logarithm.
$a^7 = 22$
the equivalent equation in logarithmic form is
(type an equation. use integers or fractions for any numbers in the equation.)
2. suppose that $f(x) = 7^{-x}$
(a) what is $f(4)$? when $x = 4$, what is the point on the graph of $f$?
(b) if $f(x) = \frac{1}{49}$, what is $x$? when $f(x) = \frac{1}{49}$, what is the point on the graph of $f$?
(a) $f(4) = $  the point $\quad$ is on the graph of $f$
(do not use commas in any individual coordinates.)
(b) if $f(x) = \frac{1}{49}, x = $  the point $\quad$ is on the graph of $f$
(do not use commas in any individual coordinates.)
3. simplify the expression
$\frac{(2a^9)^2}{(2a^9)^3}$
(simplify your answer.)
4. find the value(s) of the variable for which the rational expression is undefined
$\frac{5p}{p - 2}$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
$\boldsymbol{\circ}$ a. the expression is undefined for $p = $
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
$\boldsymbol{\circ}$ b. there are no values for which the expression is undefined.
5. use the negative exponent rules to simplify. write the answer with positive exponents only.
$\left(\frac{6}{5}\
ight)^{-3}$
$\left(\frac{6}{5}\
ight)^{-3} = $ (type an integer or a simplified fraction.)

Accepted Answer

### Question 1
# Explanation:
## Step1: Recall the exponential - logarithmic conversion rule
The exponential equation $y = a^{x}$ is equivalent to the logarithmic equation $\log_{a}y=x$. Given the exponential equation $a^{x}=22$, we can apply this rule. Here, $y = 22$, the base $a$ remains the same, and the exponent $x$ is the value of the logarithm.
## Step2: Write the logarithmic form
Using the rule $\log_{a}y = x$ for $a^{x}=22$, we get $\log_{a}22=x$

# Answer: $\log_{a}22 = x$

### Question 2 (a)
# Explanation:
## Step1: Evaluate $f(4)$ for $f(x)=7^{x}$
To find $f(4)$, we substitute $x = 4$ into the function $f(x)=7^{x}$. So $f(4)=7^{4}$
## Step2: Calculate $7^{4}$
$7^{4}=7\times7\times7\times7 = 2401$
## Step3: Find the point on the graph
For a function $y = f(x)$, when $x = 4$, $y=f(4)=2401$. So the point on the graph is $(4,2401)$

# Answer: $f(4)=2401$; The point $(4,2401)$ is on the graph of $f$

### Question 2 (b)
# Explanation:
## Step1: Solve $f(x)=\frac{1}{49}$ for $x$ when $f(x)=7^{x}$
We know that $\frac{1}{49}=\frac{1}{7^{2}}=7^{- 2}$. Since $f(x)=7^{x}$ and $f(x)=\frac{1}{49}=7^{-2}$, then $7^{x}=7^{-2}$
## Step2: Use the property of exponential functions
If $a^{m}=a^{n}$ ($a>0,a
eq1$), then $m = n$. So for $7^{x}=7^{-2}$, we have $x=-2$
## Step3: Find the point on the graph
When $x=-2$, $y = f(-2)=\frac{1}{49}$. So the point on the graph is $(-2,\frac{1}{49})$

# Answer: $x = - 2$; The point $(-2,\frac{1}{49})$ is on the graph of $f$

### Question 3
# Explanation:
## Step1: Recall the power of a product rule and power of a power rule
The power of a product rule is $(ab)^{n}=a^{n}b^{n}$ and the power of a power rule is $(a^{m})^{n}=a^{mn}$. For the expression $\frac{(2a^{9})^{2}}{(2a^{9})^{3}}$, we can also use the rule $\frac{a^{m}}{a^{n}}=a^{m - n}$ (when $a
eq0$)
## Step2: Apply the rule $\frac{a^{m}}{a^{n}}=a^{m - n}$
Let $a = 2a^{9}$, $m = 2$ and $n = 3$. Then $\frac{(2a^{9})^{2}}{(2a^{9})^{3}}=(2a^{9})^{2-3}=(2a^{9})^{-1}$
## Step3: Apply the negative - exponent rule and power of a product rule
Using the negative - exponent rule $a^{-n}=\frac{1}{a^{n}}$ and the power of a product rule $(ab)^{-n}=\frac{1}{a^{n}b^{n}}$, we have $(2a^{9})^{-1}=\frac{1}{2^{1}\times(a^{9})^{1}}$
## Step4: Simplify $(a^{9})^{1}$
Using the power of a power rule $(a^{m})^{n}=a^{mn}$, $(a^{9})^{1}=a^{9}$. So $\frac{1}{2\times a^{9}}=\frac{1}{2a^{9}}$

# Answer: $x=-2$; The point $(-2,\frac{1}{49})$ is on the graph of $f$ (Wait, correction for the simplification part: Wait, the original problem for question 3 is $\frac{(2a^{9})^{2}}{(2a^{9})^{3}}$, another way: $\frac{(2a^{9})^{2}}{(2a^{9})^{3}}=\frac{4a^{18}}{8a^{27}}=\frac{4}{8}\times\frac{a^{18}}{a^{27}}=\frac{1}{2}\times a^{18 - 27}=\frac{1}{2}a^{-9}=\frac{1}{2a^{9}}$)

### Question 4
# Explanation:
## Step1: Recall when a rational expression is undefined
A rational expression $\frac{f(p)}{g(p)}$ is undefined when the denominator $g(p)=0$. For the rational expression $\frac{5p}{p - 2}$, the denominator is $p-2$
## Step2: Solve $p - 2=0$ for $p$
Adding 2 to both sides of the equation $p - 2=0$, we get $p=2$

# Answer: A. The expression is undefined for $p = 2$

### Question 5
# Explanation:
## Step1: Recall the negative - exponent rule
The negative - exponent rule states that $(\frac{a}{b})^{-n}=(\frac{b}{a})^{n}$
## Step2: Apply the negative - exponent rule to $(\frac{6}{5})^{-3}$
Using the rule $(\frac{a}{b})^{-n}=(\frac{b}{a})^{n}$ with $a = 6$, $b = 5$ and $n = 3$, we have $(\frac{6}{5})^{-3}=(\frac{5}{6})^{3}$
## Step3: Calculate $(\frac{5}{6})^{3}$
$(\frac{5}{6})^{3}=\frac{5^{3}}{6^{3}}=\frac{125}{216}$

# Answer: $\frac{125}{216}$

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QUESTION IMAGE

Question

Explanation:

Question 1

Step1: Recall the exponential - logarithmic conversion rule

Step2: Write the logarithmic form

Step1: Evaluate \(f(4)\) for \(f(x)=7^{x}\)

Step2: Calculate \(7^{4}\)

Step3: Find the point on the graph

Step1: Solve \(f(x)=\frac{1}{49}\) for \(x\) when \(f(x)=7^{x}\)

Step2: Use the property of exponential functions

Step3: Find the point on the graph

Answer:

Question 2 (a)