Algebra Examples

$\ln (2 y - 5) + \ln (2) = 5 x + \ln (5 x)$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln ((2 y - 5) \cdot 2) = 5 x + \ln (5 x)$

Step 1.2

Apply the distributive property.

$\ln (2 y \cdot 2 - 5 \cdot 2) = 5 x + \ln (5 x)$

Step 1.3

Multiply.

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Step 1.3.1

Multiply $2$ by $2$ .

$\ln (4 y - 5 \cdot 2) = 5 x + \ln (5 x)$

Step 1.3.2

Multiply $- 5$ by $2$ .

$\ln (4 y - 10) = 5 x + \ln (5 x)$

Step 2

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

$\ln (4 y - 10) - \ln (5 x) = 5 x$

Step 3

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

$\ln (\frac{4 y - 10}{5 x}) = 5 x$

Step 4

Factor $2$ out of $4 y - 10$ .

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Step 4.1

Factor $2$ out of $4 y$ .

$\ln (\frac{2 (2 y) - 10}{5 x}) = 5 x$

Step 4.2

Factor $2$ out of $- 10$ .

$\ln (\frac{2 (2 y) + 2 (- 5)}{5 x}) = 5 x$

Step 4.3

Factor $2$ out of $2 (2 y) + 2 (- 5)$ .

$\ln (\frac{2 (2 y - 5)}{5 x}) = 5 x$

Step 5

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

$e^{\ln (\frac{2 (2 y - 5)}{5 x})} = e^{5 x}$

Step 6

Rewrite $\ln (\frac{2 (2 y - 5)}{5 x}) = 5 x$ 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 $b^{y} = x$ .

$e^{5 x} = \frac{2 (2 y - 5)}{5 x}$

Step 7

Solve for $y$ .

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Step 7.1

Rewrite the equation as $\frac{2 (2 y - 5)}{5 x} = e^{5 x}$ .

$\frac{2 (2 y - 5)}{5 x} = e^{5 x}$

Step 7.2

Multiply both sides by $5 x$ .

$\frac{2 (2 y - 5)}{5 x} (5 x) = e^{5 x} (5 x)$

Step 7.3

Simplify.

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Step 7.3.1

Simplify the left side.

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Step 7.3.1.1

Simplify $\frac{2 (2 y - 5)}{5 x} (5 x)$ .

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Step 7.3.1.1.1

Rewrite using the commutative property of multiplication.

$5 \frac{2 (2 y - 5)}{5 x} x = e^{5 x} (5 x)$

Step 7.3.1.1.2

Cancel the common factor of $5$ .

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Step 7.3.1.1.2.1

Factor $5$ out of $5 x$ .

$5 \frac{2 (2 y - 5)}{5 (x)} x = e^{5 x} (5 x)$

Step 7.3.1.1.2.2

Cancel the common factor.

$5 \frac{2 (2 y - 5)}{5 x} x = e^{5 x} (5 x)$

Step 7.3.1.1.2.3

Rewrite the expression.

$\frac{2 (2 y - 5)}{x} x = e^{5 x} (5 x)$

Step 7.3.1.1.3

Cancel the common factor of $x$ .

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Step 7.3.1.1.3.1

Cancel the common factor.

$\frac{2 (2 y - 5)}{x} x = e^{5 x} (5 x)$

Step 7.3.1.1.3.2

Rewrite the expression.

$2 (2 y - 5) = e^{5 x} (5 x)$

Step 7.3.1.1.4

Apply the distributive property.

$2 (2 y) + 2 \cdot - 5 = e^{5 x} (5 x)$

Step 7.3.1.1.5

Multiply.

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Step 7.3.1.1.5.1

Multiply $2$ by $2$ .

$4 y + 2 \cdot - 5 = e^{5 x} (5 x)$

Step 7.3.1.1.5.2

Multiply $2$ by $- 5$ .

$4 y - 10 = e^{5 x} (5 x)$

Step 7.3.2

Simplify the right side.

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Step 7.3.2.1

Simplify $e^{5 x} (5 x)$ .

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Step 7.3.2.1.1

Rewrite using the commutative property of multiplication.

$4 y - 10 = 5 e^{5 x} x$

Step 7.3.2.1.2

Reorder factors in $5 e^{5 x} x$ .

$4 y - 10 = 5 x e^{5 x}$

Step 7.4

Solve for $y$ .

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Step 7.4.1

Add $10$ to both sides of the equation.

$4 y = 5 x e^{5 x} + 10$

Step 7.4.2

Divide each term in $4 y = 5 x e^{5 x} + 10$ by $4$ and simplify.

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Step 7.4.2.1

Divide each term in $4 y = 5 x e^{5 x} + 10$ by $4$ .

$\frac{4 y}{4} = \frac{5 x e^{5 x}}{4} + \frac{10}{4}$

Step 7.4.2.2

Simplify the left side.

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Step 7.4.2.2.1

Cancel the common factor of $4$ .

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Step 7.4.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{5 x e^{5 x}}{4} + \frac{10}{4}$

Step 7.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 x e^{5 x}}{4} + \frac{10}{4}$

Step 7.4.2.3

Simplify the right side.

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Step 7.4.2.3.1

Cancel the common factor of $10$ and $4$ .

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Step 7.4.2.3.1.1

Factor $2$ out of $10$ .

$y = \frac{5 x e^{5 x}}{4} + \frac{2 (5)}{4}$

Step 7.4.2.3.1.2

Cancel the common factors.

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Step 7.4.2.3.1.2.1

Factor $2$ out of $4$ .

$y = \frac{5 x e^{5 x}}{4} + \frac{2 \cdot 5}{2 \cdot 2}$

Step 7.4.2.3.1.2.2

Cancel the common factor.

$y = \frac{5 x e^{5 x}}{4} + \frac{2 \cdot 5}{2 \cdot 2}$

Step 7.4.2.3.1.2.3

Rewrite the expression.

$y = \frac{5 x e^{5 x}}{4} + \frac{5}{2}$