Precalculus Examples

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

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

Step 1

Rewrite

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

$\log_{0.5} ({(3 x + 1)}^{\frac{1}{3}}) = - 2$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b \neq 1

$b \neq 1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

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

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

Step 2

Solve for

x

$x$ .

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Step 2.1

Rewrite the equation as

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

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

{(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

$3$ to eliminate the fractional exponent on the left side.

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

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

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}

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

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Step 2.3.1.1.1

Multiply the exponents in

{({(3 x + 1)}^{\frac{1}{3}})}^{3}

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

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Step 2.3.1.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.3.1.1.1.2

Cancel the common factor of

3

$3$ .

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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

$3$ .

$x = 21$