Precalculus Examples

$5^{x - 2} = 3^{3 x + 2}$

Step 1

Take the log of both sides of the equation.

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

Step 2

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

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

Step 3

Expand $\ln (3^{3 x + 2})$ by moving $3 x + 2$ outside the logarithm.

$(x - 2) \ln (5) = (3 x + 2) \ln (3)$

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.

$0 + 0 + (x - 2) \ln (5) = (3 x + 2) \ln (3)$

Step 4.1.2

Simplify by adding zeros.

$(x - 2) \ln (5) = (3 x + 2) \ln (3)$

Step 4.1.3

Apply the distributive property.

$x \ln (5) - 2 \ln (5) = (3 x + 2) \ln (3)$

Step 4.2

Apply the distributive property.

$x \ln (5) - 2 \ln (5) = 3 x \ln (3) + 2 \ln (3)$

Step 4.3

Subtract $3 x \ln (3)$ from both sides of the equation.

$x \ln (5) - 2 \ln (5) - 3 x \ln (3) = 2 \ln (3)$

Step 4.4

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

$x \ln (5) - 3 x \ln (3) = 2 \ln (3) + 2 \ln (5)$

Step 4.5

Factor $x$ out of $x \ln (5) - 3 x \ln (3)$ .

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Step 4.5.1

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

$x (\ln (5)) - 3 x \ln (3) = 2 \ln (3) + 2 \ln (5)$

Step 4.5.2

Factor $x$ out of $- 3 x \ln (3)$ .

$x (\ln (5)) + x (- 3 \ln (3)) = 2 \ln (3) + 2 \ln (5)$

Step 4.5.3

Factor $x$ out of $x (\ln (5)) + x (- 3 \ln (3))$ .

$x (\ln (5) - 3 \ln (3)) = 2 \ln (3) + 2 \ln (5)$

Step 4.6

Divide each term in $x (\ln (5) - 3 \ln (3)) = 2 \ln (3) + 2 \ln (5)$ by $\ln (5) - 3 \ln (3)$ and simplify.

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Step 4.6.1

Divide each term in $x (\ln (5) - 3 \ln (3)) = 2 \ln (3) + 2 \ln (5)$ by $\ln (5) - 3 \ln (3)$ .

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

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $\ln (5) - 3 \ln (3)$ .

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Step 4.6.2.1.1

Cancel the common factor.

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

Step 4.6.2.1.2

Divide $x$ by $1$ .

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

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Combine the numerators over the common denominator.

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

Step 4.6.3.2

Factor $2$ out of $2 \ln (3) + 2 \ln (5)$ .

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Step 4.6.3.2.1

Factor $2$ out of $2 \ln (3)$ .

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

Step 4.6.3.2.2

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

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

Step 4.6.3.2.3

Factor $2$ out of $2 \ln (3) + 2 \ln (5)$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 3.21163648 \dots$