Precalculus Examples

$\log_{27} (4 x - 3) = \frac{1}{3}$

Step 1

Rewrite $\log_{27} (4 x - 3) = \frac{1}{3}$ 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 $b^{y} = x$ .

$27^{\frac{1}{3}} = 4 x - 3$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $4 x - 3 = 27^{\frac{1}{3}}$ .

$4 x - 3 = 27^{\frac{1}{3}}$

Step 2.2

Simplify $27^{\frac{1}{3}}$ .

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Step 2.2.1

Rewrite.

$4 x - 3 = 0 + 0 + 27^{\frac{1}{3}}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Rewrite $27$ as $3^{3}$ .

$4 x - 3 = {(3^{3})}^{\frac{1}{3}}$

Step 2.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 x - 3 = 3^{3 (\frac{1}{3})}$

Step 2.2.3

Cancel the common factor of $3$ .

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Step 2.2.3.1

Cancel the common factor.

$4 x - 3 = 3^{3 (\frac{1}{3})}$

Step 2.2.3.2

Rewrite the expression.

$4 x - 3 = 3^{1}$

Step 2.2.4

Evaluate the exponent.

$4 x - 3 = 3$

Step 2.3

Move all terms not containing $x$ to the right side of the equation.

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Step 2.3.1

Add $3$ to both sides of the equation.

$4 x = 3 + 3$

Step 2.3.2

Add $3$ and $3$ .

$4 x = 6$

Step 2.4

Divide each term in $4 x = 6$ by $4$ and simplify.

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Step 2.4.1

Divide each term in $4 x = 6$ by $4$ .

$\frac{4 x}{4} = \frac{6}{4}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{6}{4}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{4}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Cancel the common factor of $6$ and $4$ .

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Step 2.4.3.1.1

Factor $2$ out of $6$ .

$x = \frac{2 (3)}{4}$

Step 2.4.3.1.2

Cancel the common factors.

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Step 2.4.3.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 3}{2 \cdot 2}$

Step 2.4.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 3}{2 \cdot 2}$

Step 2.4.3.1.2.3

Rewrite the expression.

$x = \frac{3}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{2}$

Decimal Form:

$x = 1.5$

Mixed Number Form:

$x = 1 \frac{1}{2}$