Precalculus Examples

$9 \log (2 x - 1) = 3$

Step 1

Divide each term in $9 \log (2 x - 1) = 3$ by $9$ and simplify.

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Step 1.1

Divide each term in $9 \log (2 x - 1) = 3$ by $9$ .

$\frac{9 \log (2 x - 1)}{9} = \frac{3}{9}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $9$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{9 \log (2 x - 1)}{9} = \frac{3}{9}$

Step 1.2.1.2

Divide $\log (2 x - 1)$ by $1$ .

$\log (2 x - 1) = \frac{3}{9}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $3$ and $9$ .

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Step 1.3.1.1

Factor $3$ out of $3$ .

$\log (2 x - 1) = \frac{3 (1)}{9}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $3$ out of $9$ .

$\log (2 x - 1) = \frac{3 \cdot 1}{3 \cdot 3}$

Step 1.3.1.2.2

Cancel the common factor.

$\log (2 x - 1) = \frac{3 \cdot 1}{3 \cdot 3}$

Step 1.3.1.2.3

Rewrite the expression.

$\log (2 x - 1) = \frac{1}{3}$

Step 2

Rewrite $\log (2 x - 1) = \frac{1}{3}$ 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 $b^{y} = x$ .

$10^{\frac{1}{3}} = 2 x - 1$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $2 x - 1 = 10^{\frac{1}{3}}$ .

$2 x - 1 = 10^{\frac{1}{3}}$

Step 3.2

Add $1$ to both sides of the equation.

$2 x = 10^{\frac{1}{3}} + 1$

Step 3.3

Divide each term in $2 x = 10^{\frac{1}{3}} + 1$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x = 10^{\frac{1}{3}} + 1$ by $2$ .

$\frac{2 x}{2} = \frac{10^{\frac{1}{3}}}{2} + \frac{1}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{10^{\frac{1}{3}}}{2} + \frac{1}{2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{10^{\frac{1}{3}}}{2} + \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{10^{\frac{1}{3}}}{2} + \frac{1}{2}$

Decimal Form:

$x = 1.57721734 \dots$