Algebra Examples

${(\frac{1}{3})}^{2 x - 5} = 120$

Step 1

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

$\ln ({(\frac{1}{3})}^{2 x - 5}) = \ln (120)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{1}{3})}^{2 x - 5})$ by moving $2 x - 5$ outside the logarithm.

$(2 x - 5) \ln (\frac{1}{3}) = \ln (120)$

Step 2.2

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

$(2 x - 5) (\ln (1) - \ln (3)) = \ln (120)$

Step 2.3

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

$(2 x - 5) (0 - \ln (3)) = \ln (120)$

Step 2.4

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

$(2 x - 5) (- \ln (3)) = \ln (120)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(2 x - 5) (- \ln (3))$ .

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Step 3.1.1

Apply the distributive property.

$2 x (- \ln (3)) - 5 (- \ln (3)) = \ln (120)$

Step 3.1.2

Multiply.

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Step 3.1.2.1

Multiply $- 1$ by $2$ .

$- 2 x \ln (3) - 5 (- \ln (3)) = \ln (120)$

Step 3.1.2.2

Multiply $- 1$ by $- 5$ .

$- 2 x \ln (3) + 5 \ln (3) = \ln (120)$

Step 4

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

$- 2 x \ln (3) + 5 \ln (3) - \ln (120) = 0$

Step 5

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

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Step 5.1

Subtract $5 \ln (3)$ from both sides of the equation.

$- 2 x \ln (3) - \ln (120) = - 5 \ln (3)$

Step 5.2

Add $\ln (120)$ to both sides of the equation.

$- 2 x \ln (3) = - 5 \ln (3) + \ln (120)$

Step 6

Divide each term in $- 2 x \ln (3) = - 5 \ln (3) + \ln (120)$ by $- 2 \ln (3)$ and simplify.

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Step 6.1

Divide each term in $- 2 x \ln (3) = - 5 \ln (3) + \ln (120)$ by $- 2 \ln (3)$ .

$\frac{- 2 x \ln (3)}{- 2 \ln (3)} = \frac{- 5 \ln (3)}{- 2 \ln (3)} + \frac{\ln (120)}{- 2 \ln (3)}$

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.

$\frac{- 2 x \ln (3)}{- 2 \ln (3)} = \frac{- 5 \ln (3)}{- 2 \ln (3)} + \frac{\ln (120)}{- 2 \ln (3)}$

Step 6.2.1.2

Rewrite the expression.

$\frac{x \ln (3)}{\ln (3)} = \frac{- 5 \ln (3)}{- 2 \ln (3)} + \frac{\ln (120)}{- 2 \ln (3)}$

Step 6.2.2

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

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Step 6.2.2.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{- 5 \ln (3)}{- 2 \ln (3)} + \frac{\ln (120)}{- 2 \ln (3)}$

Step 6.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 5 \ln (3)}{- 2 \ln (3)} + \frac{\ln (120)}{- 2 \ln (3)}$

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.

$x = \frac{- 5 \ln (3)}{- 2 \ln (3)} + \frac{\ln (120)}{- 2 \ln (3)}$

Step 6.3.1.1.2

Rewrite the expression.

$x = \frac{- 5}{- 2} + \frac{\ln (120)}{- 2 \ln (3)}$

Step 6.3.1.2

Dividing two negative values results in a positive value.

$x = \frac{5}{2} + \frac{\ln (120)}{- 2 \ln (3)}$

Step 6.3.1.3

Move the negative in front of the fraction.

$x = \frac{5}{2} - \frac{\ln (120)}{2 \ln (3)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{2} - \frac{\ln (120)}{2 \ln (3)}$

Decimal Form:

$x = 0.32111860 \dots$