Precalculus Examples

$5^{x + 2} = 7^{x - 2}$

Step 1

Take the log of both sides of the equation.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x + 2) \ln (5)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (x + 2) \ln (5) = (x - 2) \ln (7)$

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Subtract $x \ln (7)$ from both sides of the equation.

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

Step 4.4

Subtract $2 \ln (5)$ from both sides of the equation.

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

Factor $x$ out of $x (\ln (5)) + x (- 1 \ln (7))$ .

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

Step 4.6

Rewrite $- 1 \ln (7)$ as $- \ln (7)$ .

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

Simplify the left side.

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Step 4.7.2.1

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

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Step 4.7.2.1.1

Cancel the common factor.

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

Step 4.7.2.1.2

Divide $x$ by $1$ .

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

Step 4.7.3

Simplify the right side.

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Step 4.7.3.1

Combine the numerators over the common denominator.

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

Step 4.7.3.2

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

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Step 4.7.3.2.1

Factor $2$ out of $- 2 \ln (7)$ .

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

Step 4.7.3.2.2

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

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

Step 4.7.3.2.3

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

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

Step 4.7.3.3

Factor $- 1$ out of $- \ln (7)$ .

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

Step 4.7.3.4

Factor $- 1$ out of $- \ln (5)$ .

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

Step 4.7.3.5

Factor $- 1$ out of $- \ln (7) - \ln (5)$ .

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

Step 4.7.3.6

Simplify the expression.

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Step 4.7.3.6.1

Rewrite $- (\ln (7) + \ln (5))$ as $- 1 (\ln (7) + \ln (5))$ .

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

Step 4.7.3.6.2

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 21.13308424 \dots$