Algebra Examples

$\ln (4 x + 1) - \ln (3) = 5$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{4 x + 1}{3}) = 5$

Step 2

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{4 x + 1}{3})} = e^{5}$

Step 3

Rewrite $\ln (\frac{4 x + 1}{3}) = 5$ 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 $b^{y} = x$ .

$e^{5} = \frac{4 x + 1}{3}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{4 x + 1}{3} = e^{5}$ .

$\frac{4 x + 1}{3} = e^{5}$

Step 4.2

Multiply both sides by $3$ .

$\frac{4 x + 1}{3} \cdot 3 = e^{5} \cdot 3$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Cancel the common factor of $3$ .

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Step 4.3.1.1.1

Cancel the common factor.

$\frac{4 x + 1}{3} \cdot 3 = e^{5} \cdot 3$

Step 4.3.1.1.2

Rewrite the expression.

$4 x + 1 = e^{5} \cdot 3$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Move $3$ to the left of $e^{5}$ .

$4 x + 1 = 3 e^{5}$

Step 4.4

Solve for $x$ .

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Step 4.4.1

Subtract $1$ from both sides of the equation.

$4 x = 3 e^{5} - 1$

Step 4.4.2

Divide each term in $4 x = 3 e^{5} - 1$ by $4$ and simplify.

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Step 4.4.2.1

Divide each term in $4 x = 3 e^{5} - 1$ by $4$ .

$\frac{4 x}{4} = \frac{3 e^{5}}{4} + \frac{- 1}{4}$

Step 4.4.2.2

Simplify the left side.

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Step 4.4.2.2.1

Cancel the common factor of $4$ .

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Step 4.4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3 e^{5}}{4} + \frac{- 1}{4}$

Step 4.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 e^{5}}{4} + \frac{- 1}{4}$

Step 4.4.2.3

Simplify the right side.

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Step 4.4.2.3.1

Move the negative in front of the fraction.

$x = \frac{3 e^{5}}{4} - \frac{1}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 e^{5}}{4} - \frac{1}{4}$

Decimal Form:

$x = 111.05986932 \dots$