Precalculus Examples

$7^{\frac{x}{2}} = 5^{1 - x}$

Step 1

Take the log of both sides of the equation.

$\ln (7^{\frac{x}{2}}) = \ln (5^{1 - x})$

Step 2

Expand $\ln (7^{\frac{x}{2}})$ by moving $\frac{x}{2}$ outside the logarithm.

$\frac{x}{2} \ln (7) = \ln (5^{1 - x})$

Step 3

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

$\frac{x \ln (7)}{2} = \ln (5^{1 - x})$

Step 4

Expand $\ln (5^{1 - x})$ by moving $1 - x$ outside the logarithm.

$\frac{x \ln (7)}{2} = (1 - x) \ln (5)$

Step 5

Solve the equation for $x$ .

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Step 5.1

Multiply both sides by $2$ .

$\frac{x \ln (7)}{2} \cdot 2 = (1 - x) \ln (5) \cdot 2$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Cancel the common factor.

$\frac{x \ln (7)}{2} \cdot 2 = (1 - x) \ln (5) \cdot 2$

Step 5.2.1.1.2

Rewrite the expression.

$x \ln (7) = (1 - x) \ln (5) \cdot 2$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $(1 - x) \ln (5) \cdot 2$ .

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Step 5.2.2.1.1

Apply the distributive property.

$x \ln (7) = (1 \ln (5) - x \ln (5)) \cdot 2$

Step 5.2.2.1.2

Multiply $\ln (5)$ by $1$ .

$x \ln (7) = (\ln (5) - x \ln (5)) \cdot 2$

Step 5.2.2.1.3

Apply the distributive property.

$x \ln (7) = \ln (5) \cdot 2 - x \ln (5) \cdot 2$

Step 5.2.2.1.4

Simplify the expression.

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Step 5.2.2.1.4.1

Move $2$ to the left of $\ln (5)$ .

$x \ln (7) = 2 \cdot \ln (5) - x \ln (5) \cdot 2$

Step 5.2.2.1.4.2

Multiply $2$ by $- 1$ .

$x \ln (7) = 2 \cdot \ln (5) - 2 x \ln (5)$

Step 5.2.2.1.4.3

Reorder $2 \ln (5)$ and $- 2 x \ln (5)$ .

$x \ln (7) = - 2 x \ln (5) + 2 \ln (5)$

Step 5.3

Solve for $x$ .

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Step 5.3.1

Add $2 x \ln (5)$ to both sides of the equation.

$x \ln (7) + 2 x \ln (5) = 2 \ln (5)$

Step 5.3.2

Factor $x$ out of $x \ln (7) + 2 x \ln (5)$ .

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Step 5.3.2.1

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

$x (\ln (7)) + 2 x \ln (5) = 2 \ln (5)$

Step 5.3.2.2

Factor $x$ out of $2 x \ln (5)$ .

$x (\ln (7)) + x (2 \ln (5)) = 2 \ln (5)$

Step 5.3.2.3

Factor $x$ out of $x (\ln (7)) + x (2 \ln (5))$ .

$x (\ln (7) + 2 \ln (5)) = 2 \ln (5)$

Step 5.3.3

Divide each term in $x (\ln (7) + 2 \ln (5)) = 2 \ln (5)$ by $\ln (7) + 2 \ln (5)$ and simplify.

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Step 5.3.3.1

Divide each term in $x (\ln (7) + 2 \ln (5)) = 2 \ln (5)$ by $\ln (7) + 2 \ln (5)$ .

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

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

Cancel the common factor of $\ln (7) + 2 \ln (5)$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

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

Step 5.3.3.2.1.2

Divide $x$ by $1$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.62323508 \dots$