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Welcome to Bytelearn!
Let’s check out your problem:
Solve the
exponential equation
for
x
x
x
.
\newline
5
x
−
12
=
(
1
125
)
12
5
x
=
□
\begin{array}{l} 5^{x-12}=\left(\frac{1}{125}\right)^{\frac{12}{5}} \\ x=\square \end{array}
5
x
−
12
=
(
125
1
)
5
12
x
=
□
View step-by-step help
Home
Math Problems
Algebra 1
Compare linear and exponential growth
Full solution
Q.
Solve the exponential equation for
x
x
x
.
\newline
5
x
−
12
=
(
1
125
)
12
5
x
=
□
\begin{array}{l} 5^{x-12}=\left(\frac{1}{125}\right)^{\frac{12}{5}} \\ x=\square \end{array}
5
x
−
12
=
(
125
1
)
5
12
x
=
□
Recognize Exponential Form:
Recognize that both sides of the equation are written in exponential form.
\newline
We can rewrite
1
125
\frac{1}{125}
125
1
as
5
−
3
5^{-3}
5
−
3
since
125
125
125
is
5
3
5^3
5
3
.
Rewrite Using Property of Exponents:
Rewrite the right side of the equation using the property of exponents.
\newline
(
1
125
)
12
5
(\frac{1}{125})^{\frac{12}{5}}
(
125
1
)
5
12
can be rewritten as
(
5
−
3
)
12
5
(5^{-3})^{\frac{12}{5}}
(
5
−
3
)
5
12
.
Simplify Using Power Rule:
Simplify the right side of the equation using the power of a power rule.
\newline
(
5
−
3
)
12
5
(5^{-3})^{\frac{12}{5}}
(
5
−
3
)
5
12
simplifies to
5
−
36
5
5^{-\frac{36}{5}}
5
−
5
36
or
5
−
7.2
5^{-7.2}
5
−
7.2
.
Set Exponents Equal:
Set the exponents equal to each other since the bases are the same.
x
−
12
=
−
7.2
x - 12 = -7.2
x
−
12
=
−
7.2
Solve for x:
Solve for x by adding
12
12
12
to both sides of the equation.
\newline
x
−
12
+
12
=
−
7.2
+
12
x - 12 + 12 = -7.2 + 12
x
−
12
+
12
=
−
7.2
+
12
\newline
x
=
4.8
x = 4.8
x
=
4.8
More problems from Compare linear and exponential growth
Question
Jill bought a used motorcycle from a seller online for $1,200. The seller will charge her $5 a day to store the motorcycle at his house until she is able to pick it up. You can use a function to describe the total amount of money Jill will owe the seller if she waits
x
x
x
days to pick up the motorcycle.
\newline
Write an equation for the function. If it is linear, write it in the form
g
(
x
)
=
m
x
+
b
g(x) = mx + b
g
(
x
)
=
m
x
+
b
. If it is exponential, write it in the form
g
(
x
)
=
a
(
b
)
x
g(x) = a(b)^x
g
(
x
)
=
a
(
b
)
x
.
\newline
g
(
x
)
=
‾
g(x) = \underline{\hspace{3em}}
g
(
x
)
=
Get tutor help
Posted 1 year ago
Question
How does
f
(
t
)
=
−
2
t
−
7
f(t)=-2t-7
f
(
t
)
=
−
2
t
−
7
change over the interval from
t
=
−
7
t=-7
t
=
−
7
to
t
=
−
6
t=-6
t
=
−
6
?
\newline
Choices:
\newline
[
f(t) decreases by
2
]
\left[\text{f(t) decreases by } 2\right]
[
f(t) decreases by
2
]
\newline
[
f(t) increases by
2
]
\left[\text{f(t) increases by } 2\right]
[
f(t) increases by
2
]
\newline
[
f(t) increases by
200
%
]
\left[\text{f(t) increases by } 200\%\right]
[
f(t) increases by
200%
]
\newline
[
f(t) decreases by a factor of
2
]
\left[\text{f(t) decreases by a factor of } 2\right]
[
f(t) decreases by a factor of
2
]
Get tutor help
Posted 1 year ago
Question
How does
g
(
t
)
=
9
t
g(t)=9^t
g
(
t
)
=
9
t
change over the interval from
t
=
1
t=1
t
=
1
to
t
=
3
t=3
t
=
3
?
\newline
Choices:
\newline
[
g(t) decreases by a factor of
81
]
\left[\text{g(t) decreases by a factor of } 81\right]
[
g(t) decreases by a factor of
81
]
\newline
[
g(t) increases by a factor of
81
]
\left[\text{g(t) increases by a factor of } 81\right]
[
g(t) increases by a factor of
81
]
\newline
[
g(t) increases by
18
%
]
\left[\text{g(t) increases by } 18\%\right]
[
g(t) increases by
18%
]
\newline
[
g(t) decreases by
9
%
]
\left[\text{g(t) decreases by } 9\%\right]
[
g(t) decreases by
9%
]
Get tutor help
Posted 1 year ago
Question
Both of these functions grow as
x
x
x
gets larger and larger. Which function eventually exceeds the other?
\newline
Choices:
\newline
(
A
)
f
(
x
)
=
8
x
+
3.3
(A)f(x) = 8x + 3.3
(
A
)
f
(
x
)
=
8
x
+
3.3
\newline
(
B
)
g
(
x
)
=
3.
3
x
−
5
(B)g(x) = 3.3^x - 5
(
B
)
g
(
x
)
=
3.
3
x
−
5
Get tutor help
Posted 1 year ago
Question
Both of these functions grow as
x
x
x
gets larger and larger. Which function eventually exceeds the other?
\newline
Choices:
\newline
(
A
)
f
(
x
)
=
2
x
+
9
(A) \ f(x) = 2x + 9
(
A
)
f
(
x
)
=
2
x
+
9
\newline
(
B
)
g
(
x
)
=
2
x
(B)\ g(x) = 2^x
(
B
)
g
(
x
)
=
2
x
Get tutor help
Posted 1 year ago
Question
The functions
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
are differentiable.
\newline
The function
h
(
x
)
h(x)
h
(
x
)
is defined as:
h
(
x
)
=
g
(
x
)
f
(
x
)
h(x)=\frac{g(x)}{f(x)}
h
(
x
)
=
f
(
x
)
g
(
x
)
\newline
If
f
(
8
)
=
1
f(8)=1
f
(
8
)
=
1
,
f
′
(
8
)
=
−
6
f'(8)=-6
f
′
(
8
)
=
−
6
,
g
(
8
)
=
8
g(8)=8
g
(
8
)
=
8
, and
g
′
(
8
)
=
−
10
g'(8)=-10
g
′
(
8
)
=
−
10
, what is
h
′
(
8
)
h'(8)
h
′
(
8
)
?
\newline
Simplify any fractions.
\newline
h
′
(
8
)
=
h'(8)=
h
′
(
8
)
=
______
Get tutor help
Posted 1 year ago
Question
The functions
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
are differentiable.
\newline
The function
r
(
x
)
r(x)
r
(
x
)
is defined as:
r
(
x
)
=
p
(
x
)
q
(
x
)
r(x)= \frac{p(x)}{q(x)}
r
(
x
)
=
q
(
x
)
p
(
x
)
\newline
If
p
(
3
)
=
2
p(3)= 2
p
(
3
)
=
2
,
p
′
(
3
)
=
4
p'(3)= 4
p
′
(
3
)
=
4
,
q
(
3
)
=
6
q(3)= 6
q
(
3
)
=
6
, and
q
′
(
3
)
=
9
q'(3)= 9
q
′
(
3
)
=
9
, what is
r
′
(
3
)
r'(3)
r
′
(
3
)
?
\newline
Simplify any fractions.
\newline
r
′
(
3
)
=
r'(3)=
r
′
(
3
)
=
_____
Get tutor help
Posted 1 year ago
Question
The functions
u
(
x
)
u(x)
u
(
x
)
and
v
(
x
)
v(x)
v
(
x
)
are differentiable.
\newline
The function
w
(
x
)
w(x)
w
(
x
)
is defined as:
w
(
x
)
=
u
(
x
)
v
(
x
)
w(x)= \frac{u(x)}{v(x)}
w
(
x
)
=
v
(
x
)
u
(
x
)
\newline
If
u
(
5
)
=
3
u(5)= 3
u
(
5
)
=
3
,
u
′
(
5
)
=
−
2
u'(5)= -2
u
′
(
5
)
=
−
2
,
v
(
5
)
=
7
v(5)= 7
v
(
5
)
=
7
, and
v
′
(
5
)
=
1
v'(5)= 1
v
′
(
5
)
=
1
, what is
w
′
(
5
)
w'(5)
w
′
(
5
)
?
\newline
Simplify any fractions.
\newline
w
′
(
5
)
=
w'(5)=
w
′
(
5
)
=
_____
Get tutor help
Posted 1 year ago
Question
The functions
a
(
x
)
a(x)
a
(
x
)
and
b
(
x
)
b(x)
b
(
x
)
are differentiable.
\newline
The function
c
(
x
)
c(x)
c
(
x
)
is defined as:
c
(
x
)
=
a
(
x
)
b
(
x
)
c(x)= \frac{a(x)}{b(x)}
c
(
x
)
=
b
(
x
)
a
(
x
)
\newline
If
a
(
2
)
=
4
a(2)= 4
a
(
2
)
=
4
,
a
′
(
2
)
=
5
a'(2)= 5
a
′
(
2
)
=
5
,
b
(
2
)
=
1
b(2)= 1
b
(
2
)
=
1
, and
b
′
(
2
)
=
−
3
b'(2)= -3
b
′
(
2
)
=
−
3
, what is
c
′
(
2
)
c'(2)
c
′
(
2
)
?
\newline
Simplify any fractions.
\newline
c
′
(
2
)
=
c'(2)=
c
′
(
2
)
=
_____
Get tutor help
Posted 1 year ago
Question
The functions
m
(
x
)
m(x)
m
(
x
)
and
n
(
x
)
n(x)
n
(
x
)
are differentiable.
\newline
The function
o
(
x
)
o(x)
o
(
x
)
is defined as:
o
(
x
)
=
m
(
x
)
n
(
x
)
o(x)= \frac{m(x)}{n(x)}
o
(
x
)
=
n
(
x
)
m
(
x
)
\newline
If
m
(
7
)
=
2
m(7)= 2
m
(
7
)
=
2
,
m
′
(
7
)
=
−
1
m'(7)= -1
m
′
(
7
)
=
−
1
,
n
(
7
)
=
4
n(7)= 4
n
(
7
)
=
4
, and
n
′
(
7
)
=
8
n'(7)= 8
n
′
(
7
)
=
8
, what is
o
′
(
7
)
o'(7)
o
′
(
7
)
?
\newline
Simplify any fractions.
\newline
o
′
(
7
)
=
o'(7)=
o
′
(
7
)
=
_____
Get tutor help
Posted 1 year ago
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