Algebra Examples

$7^{x - 3} = 4 \cdot 5^{\frac{x}{2}}$

Step 1

Take the log of both sides of the equation.

$\ln (7^{x - 3}) = \ln (4 \cdot 5^{\frac{x}{2}})$

Step 2

Expand $\ln (7^{x - 3})$ by moving $x - 3$ outside the logarithm.

$(x - 3) \ln (7) = \ln (4 \cdot 5^{\frac{x}{2}})$

Step 3

Rewrite $\ln (4 \cdot 5^{\frac{x}{2}})$ as $\ln (4) + \ln (5^{\frac{x}{2}})$ .

$(x - 3) \ln (7) = \ln (4) + \ln (5^{\frac{x}{2}})$

Step 4

Expand $\ln (5^{\frac{x}{2}})$ by moving $\frac{x}{2}$ outside the logarithm.

$(x - 3) \ln (7) = \ln (4) + \frac{x}{2} \ln (5)$

Step 5

Combine $\frac{x}{2}$ and $\ln (5)$ .

$(x - 3) \ln (7) = \ln (4) + \frac{x \ln (5)}{2}$

Step 6

Solve the equation for $x$ .

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $(x - 3) \ln (7)$ .

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Step 6.1.1.1

Apply the distributive property.

$x \ln (7) - 3 \ln (7) = \ln (4) + \frac{x \ln (5)}{2}$

Step 6.1.1.2

Simplify $- 3 \ln (7)$ by moving $3$ inside the logarithm.

$x \ln (7) - \ln (7^{3}) = \ln (4) + \frac{x \ln (5)}{2}$

Step 6.1.1.3

Raise $7$ to the power of $3$ .

$x \ln (7) - \ln (343) = \ln (4) + \frac{x \ln (5)}{2}$

Step 6.2

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

$x \ln (7) - \ln (343) - \ln (4) - \frac{x \ln (5)}{2} = 0$

Step 6.3

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

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Step 6.3.1

Add $\ln (343)$ to both sides of the equation.

$x \ln (7) - \ln (4) - \frac{x \ln (5)}{2} = \ln (343)$

Step 6.3.2

Add $\ln (4)$ to both sides of the equation.

$x \ln (7) - \frac{x \ln (5)}{2} = \ln (343) + \ln (4)$

Step 6.4

Factor $x$ out of $x \ln (7) - \frac{x \ln (5)}{2}$ .

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Step 6.4.1

Factor $x$ out of $x \ln (7)$ .

$x (\ln (7)) - \frac{x \ln (5)}{2} = \ln (343) + \ln (4)$

Step 6.4.2

Factor $x$ out of $- \frac{x \ln (5)}{2}$ .

$x (\ln (7)) + x (- \frac{\ln (5)}{2}) = \ln (343) + \ln (4)$

Step 6.4.3

Factor $x$ out of $x (\ln (7)) + x (- \frac{\ln (5)}{2})$ .

$x (\ln (7) - \frac{\ln (5)}{2}) = \ln (343) + \ln (4)$

Step 6.5

Divide each term in $x (\ln (7) - \frac{\ln (5)}{2}) = \ln (343) + \ln (4)$ by $\ln (7) - \frac{\ln (5)}{2}$ and simplify.

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Step 6.5.1

Divide each term in $x (\ln (7) - \frac{\ln (5)}{2}) = \ln (343) + \ln (4)$ by $\ln (7) - \frac{\ln (5)}{2}$ .

$\frac{x (\ln (7) - \frac{\ln (5)}{2})}{\ln (7) - \frac{\ln (5)}{2}} = \frac{\ln (343)}{\ln (7) - \frac{\ln (5)}{2}} + \frac{\ln (4)}{\ln (7) - \frac{\ln (5)}{2}}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $\ln (7) - \frac{\ln (5)}{2}$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{x (\ln (7) - \frac{\ln (5)}{2})}{\ln (7) - \frac{\ln (5)}{2}} = \frac{\ln (343)}{\ln (7) - \frac{\ln (5)}{2}} + \frac{\ln (4)}{\ln (7) - \frac{\ln (5)}{2}}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (343)}{\ln (7) - \frac{\ln (5)}{2}} + \frac{\ln (4)}{\ln (7) - \frac{\ln (5)}{2}}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Simplify each term.

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Step 6.5.3.1.1

Simplify the denominator.

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Step 6.5.3.1.1.1

To write $\ln (7)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{\ln (343)}{\ln (7) \cdot \frac{2}{2} - \frac{\ln (5)}{2}} + \frac{\ln (4)}{\ln (7) - \frac{\ln (5)}{2}}$

Step 6.5.3.1.1.2

Combine $\ln (7)$ and $\frac{2}{2}$ .

$x = \frac{\ln (343)}{\frac{\ln (7) \cdot 2}{2} - \frac{\ln (5)}{2}} + \frac{\ln (4)}{\ln (7) - \frac{\ln (5)}{2}}$

Step 6.5.3.1.1.3

Combine the numerators over the common denominator.

$x = \frac{\ln (343)}{\frac{\ln (7) \cdot 2 - \ln (5)}{2}} + \frac{\ln (4)}{\ln (7) - \frac{\ln (5)}{2}}$

Step 6.5.3.1.1.4

Move $2$ to the left of $\ln (7)$ .

$x = \frac{\ln (343)}{\frac{2 \ln (7) - \ln (5)}{2}} + \frac{\ln (4)}{\ln (7) - \frac{\ln (5)}{2}}$

Step 6.5.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = \ln (343) \frac{2}{2 \ln (7) - \ln (5)} + \frac{\ln (4)}{\ln (7) - \frac{\ln (5)}{2}}$

Step 6.5.3.1.3

Combine $\ln (343)$ and $\frac{2}{2 \ln (7) - \ln (5)}$ .

$x = \frac{\ln (343) \cdot 2}{2 \ln (7) - \ln (5)} + \frac{\ln (4)}{\ln (7) - \frac{\ln (5)}{2}}$

Step 6.5.3.1.4

Move $2$ to the left of $\ln (343)$ .

$x = \frac{2 \cdot \ln (343)}{2 \ln (7) - \ln (5)} + \frac{\ln (4)}{\ln (7) - \frac{\ln (5)}{2}}$

Step 6.5.3.1.5

Simplify the denominator.

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Step 6.5.3.1.5.1

To write $\ln (7)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{2 \ln (343)}{2 \ln (7) - \ln (5)} + \frac{\ln (4)}{\ln (7) \cdot \frac{2}{2} - \frac{\ln (5)}{2}}$

Step 6.5.3.1.5.2

Combine $\ln (7)$ and $\frac{2}{2}$ .

$x = \frac{2 \ln (343)}{2 \ln (7) - \ln (5)} + \frac{\ln (4)}{\frac{\ln (7) \cdot 2}{2} - \frac{\ln (5)}{2}}$

Step 6.5.3.1.5.3

Combine the numerators over the common denominator.

$x = \frac{2 \ln (343)}{2 \ln (7) - \ln (5)} + \frac{\ln (4)}{\frac{\ln (7) \cdot 2 - \ln (5)}{2}}$

Step 6.5.3.1.5.4

Move $2$ to the left of $\ln (7)$ .

$x = \frac{2 \ln (343)}{2 \ln (7) - \ln (5)} + \frac{\ln (4)}{\frac{2 \ln (7) - \ln (5)}{2}}$

Step 6.5.3.1.6

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{2 \ln (343)}{2 \ln (7) - \ln (5)} + \ln (4) \frac{2}{2 \ln (7) - \ln (5)}$

Step 6.5.3.1.7

Combine $\ln (4)$ and $\frac{2}{2 \ln (7) - \ln (5)}$ .

$x = \frac{2 \ln (343)}{2 \ln (7) - \ln (5)} + \frac{\ln (4) \cdot 2}{2 \ln (7) - \ln (5)}$

Step 6.5.3.1.8

Move $2$ to the left of $\ln (4)$ .

$x = \frac{2 \ln (343)}{2 \ln (7) - \ln (5)} + \frac{2 \ln (4)}{2 \ln (7) - \ln (5)}$

Step 6.5.3.2

Simplify terms.

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Step 6.5.3.2.1

Combine the numerators over the common denominator.

$x = \frac{2 \ln (343) + 2 \ln (4)}{2 \ln (7) - \ln (5)}$

Step 6.5.3.2.2

Factor $2$ out of $2 \ln (343) + 2 \ln (4)$ .

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Step 6.5.3.2.2.1

Factor $2$ out of $2 \ln (343)$ .

$x = \frac{2 \ln (343) + 2 \ln (4)}{2 \ln (7) - \ln (5)}$

Step 6.5.3.2.2.2

Factor $2$ out of $2 \ln (4)$ .

$x = \frac{2 \ln (343) + 2 \ln (4)}{2 \ln (7) - \ln (5)}$

Step 6.5.3.2.2.3

Factor $2$ out of $2 \ln (343) + 2 \ln (4)$ .

$x = \frac{2 (\ln (343) + \ln (4))}{2 \ln (7) - \ln (5)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 (\ln (343) + \ln (4))}{2 \ln (7) - \ln (5)}$

Decimal Form:

$x = 6.33024935 \dots$