Finite Math Examples

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

Step 1

Take the log of both sides of the equation.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x + 3) \ln (3)$ .

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

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

$x (\ln (3)) + x (- 1 \ln (2)) = - 3 \ln (3)$

Step 4.4.3

Factor $x$ out of $x (\ln (3)) + x (- 1 \ln (2))$ .

$x (\ln (3) - 1 \ln (2)) = - 3 \ln (3)$

Step 4.5

Rewrite $- 1 \ln (2)$ as $- \ln (2)$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $\ln (3) - \ln (2)$ .

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Step 4.6.2.1.1

Cancel the common factor.

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

Step 4.6.2.1.2

Divide $x$ by $1$ .

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

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 8.12853387 \dots$