Precalculus Examples

${\log_{6}}^{2} (x) = \log_{6} (x)$

Step 1

Remove parentheses.

${\log_{6}}^{2} (x) = \log_{6} (x)$

Step 2

Subtract $\log_{6} (x)$ from both sides of the equation.

${\log_{6}}^{2} (x) - \log_{6} (x) = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Let $u = \log_{6} (x)$ . Substitute $u$ for all occurrences of $\log_{6} (x)$ .

$u^{2} - u = 0$

Step 3.2

Factor $u$ out of $u^{2} - u$ .

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Step 3.2.1

Factor $u$ out of $u^{2}$ .

$u \cdot u - u = 0$

Step 3.2.2

Factor $u$ out of $- u$ .

$u \cdot u + u \cdot - 1 = 0$

Step 3.2.3

Factor $u$ out of $u \cdot u + u \cdot - 1$ .

$u (u - 1) = 0$

Step 3.3

Replace all occurrences of $u$ with $\log_{6} (x)$ .

$\log_{6} (x) (\log_{6} (x) - 1) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\log_{6} (x) = 0$

$\log_{6} (x) - 1 = 0$

Step 5

Set $\log_{6} (x)$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $\log_{6} (x)$ equal to $0$ .

$\log_{6} (x) = 0$

Step 5.2

Solve $\log_{6} (x) = 0$ for $x$ .

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Step 5.2.1

Rewrite $\log_{6} (x) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$6^{0} = x$

Step 5.2.2

Solve for $x$ .

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Step 5.2.2.1

Rewrite the equation as $x = 6^{0}$ .

$x = 6^{0}$

Step 5.2.2.2

Anything raised to $0$ is $1$ .

$x = 1$

Step 6

Set $\log_{6} (x) - 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $\log_{6} (x) - 1$ equal to $0$ .

$\log_{6} (x) - 1 = 0$

Step 6.2

Solve $\log_{6} (x) - 1 = 0$ for $x$ .

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Step 6.2.1

Add $1$ to both sides of the equation.

$\log_{6} (x) = 1$

Step 6.2.2

Rewrite $\log_{6} (x) = 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$6^{1} = x$

Step 6.2.3

Rewrite the equation as $x = 6^{1}$ .

$x = 6$

Step 7

The final solution is all the values that make $\log_{6} (x) (\log_{6} (x) - 1) = 0$ true.

$x = 1, 6$