Question Solve for $x$ in the equation $7^{x} = \frac{1}{7}$ . Express the answer as an integer or a simplified fraction.

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $7^{x} = \frac{1}{7}$ .
2. We need to solve for $x$ .
3. We will use the properties of exponents to solve the equation.

STEP 2

Recognize that $\frac{1}{7}$ can be written as $7^{-1}$ , because any non-zero number to the negative power is the reciprocal of the number to the corresponding positive power.
$\frac{1}{7} = 7^{-1}$

STEP 3

Now we can rewrite the original equation using this property.
$7^{x} = 7^{-1}$

STEP 4

Since the bases on both sides of the equation are the same (both are 7), we can set the exponents equal to each other. This is because if $a^m = a^n$ for some non-zero number $a$ , then $m = n$ .
$x = -1$

STEP 5

We have found the value of $x$ .
$x = -1$
The solution to the equation $7^{x} = \frac{1}{7}$ is $x = -1$ .

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Question Solve for $x$ in the equation $7^{x} = \frac{1}{7}$ . Express the answer as an integer or a simplified fraction.

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $7^{x} = \frac{1}{7}$ .
2. We need to solve for $x$ .
3. We will use the properties of exponents to solve the equation.

STEP 2

Recognize that $\frac{1}{7}$ can be written as $7^{-1}$ , because any non-zero number to the negative power is the reciprocal of the number to the corresponding positive power.
$\frac{1}{7} = 7^{-1}$

STEP 3

Now we can rewrite the original equation using this property.
$7^{x} = 7^{-1}$

STEP 4

Since the bases on both sides of the equation are the same (both are 7), we can set the exponents equal to each other. This is because if $a^m = a^n$ for some non-zero number $a$ , then $m = n$ .
$x = -1$

STEP 5

We have found the value of $x$ .
$x = -1$
The solution to the equation $7^{x} = \frac{1}{7}$ is $x = -1$ .

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Question Solve for xxx in the equation 7x=177^{x} = \frac{1}{7}7x=71​. Express the answer as an integer or a simplified fraction.

Studdy Solution

STEP 1

Assumptions1. We are given the equation 7x=177^{x} = \frac{1}{7}7x=71​.2. We need to solve for xxx.3. We will use the properties of exponents to solve the equation.

STEP 2

Recognize that 17\frac{1}{7}71​ can be written as 7−17^{-1}7−1, because any non-zero number to the negative power is the reciprocal of the number to the corresponding positive power.17=7−1\frac{1}{7} = 7^{-1}71​=7−1

STEP 3

Now we can rewrite the original equation using this property.7x=7−17^{x} = 7^{-1}7x=7−1

STEP 4

Since the bases on both sides of the equation are the same (both are 7), we can set the exponents equal to each other. This is because if am=ana^m = a^nam=an for some non-zero number aaa, then m=nm = nm=n.x=−1x = -1x=−1

STEP 5

We have found the value of xxx.x=−1x = -1x=−1The solution to the equation 7x=177^{x} = \frac{1}{7}7x=71​ is x=−1x = -1x=−1.

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Question Solve for $x$ in the equation $7^{x} = \frac{1}{7}$ . Express the answer as an integer or a simplified fraction.

Assumptions
1. We are given the equation $7^{x} = \frac{1}{7}$ .
2. We need to solve for $x$ .
3. We will use the properties of exponents to solve the equation.

Recognize that $\frac{1}{7}$ can be written as $7^{-1}$ , because any non-zero number to the negative power is the reciprocal of the number to the corresponding positive power.
$\frac{1}{7} = 7^{-1}$

Now we can rewrite the original equation using this property.
$7^{x} = 7^{-1}$

Since the bases on both sides of the equation are the same (both are 7), we can set the exponents equal to each other. This is because if $a^m = a^n$ for some non-zero number $a$ , then $m = n$ .
$x = -1$

We have found the value of $x$ .
$x = -1$
The solution to the equation $7^{x} = \frac{1}{7}$ is $x = -1$ .