Math / Algebra

QuestionSolve for $x$ : $\begin{array}{l} \log _{3}\left(x^{7}\right)=-1 \\ x= \end{array}$

Studdy Solution

STEP 1

1. The equation involves a logarithm with base 3.
2. We will need to use properties of logarithms and exponents to solve for $x$ .

STEP 2

1. Use the definition of logarithms to rewrite the equation in exponential form.
2. Solve the resulting equation for $x$ .

STEP 3

Use the definition of a logarithm to rewrite the equation in exponential form. The equation $\log_{3}(x^7) = -1$ implies:
$3^{-1} = x^7$

STEP 4

Solve the exponential equation for $x$ . First, recognize that $3^{-1}$ is the same as $\frac{1}{3}$ :
$x^7 = \frac{1}{3}$

STEP 5

To solve for $x$ , take the seventh root of both sides. This can be expressed as raising both sides to the power of $\frac{1}{7}$ :
$x = \left(\frac{1}{3}\right)^{\frac{1}{7}}$
The solution for $x$ is:
$\boxed{\left(\frac{1}{3}\right)^{\frac{1}{7}}}$

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QuestionSolve for $x$ : $\begin{array}{l} \log _{3}\left(x^{7}\right)=-1 \\ x= \end{array}$

Studdy Solution

STEP 1

1. The equation involves a logarithm with base 3.
2. We will need to use properties of logarithms and exponents to solve for $x$ .

STEP 2

1. Use the definition of logarithms to rewrite the equation in exponential form.
2. Solve the resulting equation for $x$ .

STEP 3

Use the definition of a logarithm to rewrite the equation in exponential form. The equation $\log_{3}(x^7) = -1$ implies:
$3^{-1} = x^7$

STEP 4

Solve the exponential equation for $x$ . First, recognize that $3^{-1}$ is the same as $\frac{1}{3}$ :
$x^7 = \frac{1}{3}$

STEP 5

To solve for $x$ , take the seventh root of both sides. This can be expressed as raising both sides to the power of $\frac{1}{7}$ :
$x = \left(\frac{1}{3}\right)^{\frac{1}{7}}$
The solution for $x$ is:
$\boxed{\left(\frac{1}{3}\right)^{\frac{1}{7}}}$

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QuestionSolve for xxx : log⁡3(x7)=−1x=\begin{array}{l} \log _{3}\left(x^{7}\right)=-1 \\ x= \end{array}log3​(x7)=−1x=​

Studdy Solution

STEP 1

1. The equation involves a logarithm with base 3.2. We will need to use properties of logarithms and exponents to solve for x x x.

STEP 2

1. Use the definition of logarithms to rewrite the equation in exponential form.2. Solve the resulting equation for x x x.

STEP 3

Use the definition of a logarithm to rewrite the equation in exponential form. The equation log⁡3(x7)=−1\log_{3}(x^7) = -1log3​(x7)=−1 implies:3−1=x7 3^{-1} = x^7 3−1=x7

STEP 4

Solve the exponential equation for x x x. First, recognize that 3−1 3^{-1} 3−1 is the same as 13\frac{1}{3}31​:x7=13 x^7 = \frac{1}{3} x7=31​

STEP 5

To solve for x x x, take the seventh root of both sides. This can be expressed as raising both sides to the power of 17\frac{1}{7}71​:x=(13)17 x = \left(\frac{1}{3}\right)^{\frac{1}{7}} x=(31​)71​ The solution for x x x is:(13)17 \boxed{\left(\frac{1}{3}\right)^{\frac{1}{7}}} (31​)71​​

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionSolve for $x$ : $\begin{array}{l} \log _{3}\left(x^{7}\right)=-1 \\ x= \end{array}$

1. The equation involves a logarithm with base 3.
2. We will need to use properties of logarithms and exponents to solve for $x$ .

1. Use the definition of logarithms to rewrite the equation in exponential form.
2. Solve the resulting equation for $x$ .

Use the definition of a logarithm to rewrite the equation in exponential form. The equation $\log_{3}(x^7) = -1$ implies:
$3^{-1} = x^7$

Solve the exponential equation for $x$ . First, recognize that $3^{-1}$ is the same as $\frac{1}{3}$ :
$x^7 = \frac{1}{3}$

To solve for $x$ , take the seventh root of both sides. This can be expressed as raising both sides to the power of $\frac{1}{7}$ :
$x = \left(\frac{1}{3}\right)^{\frac{1}{7}}$
The solution for $x$ is:
$\boxed{\left(\frac{1}{3}\right)^{\frac{1}{7}}}$