Precalculus Examples

$(3 x - 5) = \ln (11) + \ln (2)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\ln (11) + \ln (2)$ .

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Step 1.1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$3 x - 5 = \ln (11 \cdot 2)$

Step 1.1.2

Multiply $11$ by $2$ .

$3 x - 5 = \ln (22)$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$- \ln (22) = - 3 x + 5$

Step 3

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 3 x + 5 = - \ln (22)$

Step 4

Subtract $5$ from both sides of the equation.

$- 3 x = - \ln (22) - 5$

Step 5

Divide each term in $- 3 x = - \ln (22) - 5$ by $- 3$ and simplify.

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Step 5.1

Divide each term in $- 3 x = - \ln (22) - 5$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- \ln (22)}{- 3} + \frac{- 5}{- 3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- \ln (22)}{- 3} + \frac{- 5}{- 3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (22)}{- 3} + \frac{- 5}{- 3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Dividing two negative values results in a positive value.

$x = \frac{\ln (22)}{3} + \frac{- 5}{- 3}$

Step 5.3.1.2

Rewrite $\frac{\ln (22)}{3}$ as $\frac{1}{3} \ln (22)$ .

$x = \frac{1}{3} \ln (22) + \frac{- 5}{- 3}$

Step 5.3.1.3

Simplify $\frac{1}{3} \ln (22)$ by moving $\frac{1}{3}$ inside the logarithm.

$x = \ln (22^{\frac{1}{3}}) + \frac{- 5}{- 3}$

Step 5.3.1.4

Dividing two negative values results in a positive value.

$x = \ln (22^{\frac{1}{3}}) + \frac{5}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \ln (22^{\frac{1}{3}}) + \frac{5}{3}$

Decimal Form:

$x = 2.69701415 \dots$