Precalculus Examples

$\log_{0.5} ({(3 x + 1)}^{\frac{1}{3}}) = - 2$

Step 1

Rewrite $\log_{0.5} ({(3 x + 1)}^{\frac{1}{3}}) = - 2$ 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 $b^{y} = x$ .

${0.5}^{- 2} = {(3 x + 1)}^{\frac{1}{3}}$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as ${(3 x + 1)}^{\frac{1}{3}} = {0.5}^{- 2}$ .

${(3 x + 1)}^{\frac{1}{3}} = {0.5}^{- 2}$

Step 2.2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({(3 x + 1)}^{\frac{1}{3}})}^{3} = {({0.5}^{- 2})}^{3}$

Step 2.3

Simplify the exponent.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify ${({(3 x + 1)}^{\frac{1}{3}})}^{3}$ .

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Step 2.3.1.1.1

Multiply the exponents in ${({(3 x + 1)}^{\frac{1}{3}})}^{3}$ .

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Step 2.3.1.1.1.1

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

${(3 x + 1)}^{\frac{1}{3} \cdot 3} = {({0.5}^{- 2})}^{3}$

Step 2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.3.1.1.1.2.1

Cancel the common factor.

${(3 x + 1)}^{\frac{1}{3} \cdot 3} = {({0.5}^{- 2})}^{3}$

Step 2.3.1.1.1.2.2

Rewrite the expression.

${(3 x + 1)}^{1} = {({0.5}^{- 2})}^{3}$

Step 2.3.1.1.2

Simplify.

$3 x + 1 = {({0.5}^{- 2})}^{3}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify ${({0.5}^{- 2})}^{3}$ .

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Step 2.3.2.1.1

Multiply the exponents in ${({0.5}^{- 2})}^{3}$ .

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Step 2.3.2.1.1.1

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

$3 x + 1 = {0.5}^{- 2 \cdot 3}$

Step 2.3.2.1.1.2

Multiply $- 2$ by $3$ .

$3 x + 1 = {0.5}^{- 6}$

Step 2.3.2.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 x + 1 = \frac{1}{{0.5}^{6}}$

Step 2.3.2.1.3

Raise $0.5$ to the power of $6$ .

$3 x + 1 = \frac{1}{0.015625}$

Step 2.3.2.1.4

Divide $1$ by $0.015625$ .

$3 x + 1 = 64$

Step 2.4

Solve for $x$ .

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Step 2.4.1

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

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Step 2.4.1.1

Subtract $1$ from both sides of the equation.

$3 x = 64 - 1$

Step 2.4.1.2

Subtract $1$ from $64$ .

$3 x = 63$

Step 2.4.2

Divide each term in $3 x = 63$ by $3$ and simplify.

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Step 2.4.2.1

Divide each term in $3 x = 63$ by $3$ .

$\frac{3 x}{3} = \frac{63}{3}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{63}{3}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{63}{3}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $63$ by $3$ .

$x = 21$