Algebra Examples

${(\frac{1}{3})}^{2 x + 3} = 5$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({(\frac{1}{3})}^{2 x + 3}) = \ln (5)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{1}{3})}^{2 x + 3})$ by moving $2 x + 3$ outside the logarithm.

$(2 x + 3) \ln (\frac{1}{3}) = \ln (5)$

Step 2.2

Rewrite $\ln (\frac{1}{3})$ as $\ln (1) - \ln (3)$ .

$(2 x + 3) (\ln (1) - \ln (3)) = \ln (5)$

Step 2.3

The natural logarithm of $1$ is $0$ .

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

Step 2.4

Subtract $\ln (3)$ from $0$ .

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

Step 3

Simplify the left side.

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Step 3.1

Simplify $(2 x + 3) (- \ln (3))$ .

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Multiply.

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Step 3.1.2.1

Multiply $- 1$ by $2$ .

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

Step 3.1.2.2

Multiply $- 1$ by $3$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

Add $3 \ln (3)$ to both sides of the equation.

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

Step 5.2

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 2$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Rewrite the expression.

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

Step 6.2.2

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

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Step 6.2.2.1

Cancel the common factor.

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

Step 6.2.2.2

Divide $x$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

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Step 6.3.1.1.1

Cancel the common factor.

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

Step 6.3.1.1.2

Rewrite the expression.

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

Step 6.3.1.2

Move the negative in front of the fraction.

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

Step 6.3.1.3

Move the negative in front of the fraction.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 2.23248676 \dots$