Algebra Examples

$5 \cdot 17^{1 - 3 x} = 20$

Step 1

Divide each term in $5 \cdot 17^{1 - 3 x} = 20$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 \cdot 17^{1 - 3 x} = 20$ by $5$ .

$\frac{5 \cdot 17^{1 - 3 x}}{5} = \frac{20}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 \cdot 17^{1 - 3 x}}{5} = \frac{20}{5}$

Step 1.2.1.2

Divide $17^{1 - 3 x}$ by $1$ .

$17^{1 - 3 x} = \frac{20}{5}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $20$ by $5$ .

$17^{1 - 3 x} = 4$

Step 2

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

$\ln (17^{1 - 3 x}) = \ln (4)$

Step 3

Expand $\ln (17^{1 - 3 x})$ by moving $1 - 3 x$ outside the logarithm.

$(1 - 3 x) \ln (17) = \ln (4)$

Step 4

Simplify the left side.

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Step 4.1

Simplify $(1 - 3 x) \ln (17)$ .

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Step 4.1.1

Apply the distributive property.

$1 \ln (17) - 3 x \ln (17) = \ln (4)$

Step 4.1.2

Multiply $\ln (17)$ by $1$ .

$\ln (17) - 3 x \ln (17) = \ln (4)$

Step 5

Reorder $\ln (17)$ and $- 3 x \ln (17)$ .

$- 3 x \ln (17) + \ln (17) = \ln (4)$

Step 6

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

$- 3 x \ln (17) + \ln (17) - \ln (4) = 0$

Step 7

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- 3 x \ln (17) + \ln (\frac{17}{4}) = 0$

Step 8

Subtract $\ln (\frac{17}{4})$ from both sides of the equation.

$- 3 x \ln (17) = - \ln (\frac{17}{4})$

Step 9

Divide each term in $- 3 x \ln (17) = - \ln (\frac{17}{4})$ by $- 3 \ln (17)$ and simplify.

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Step 9.1

Divide each term in $- 3 x \ln (17) = - \ln (\frac{17}{4})$ by $- 3 \ln (17)$ .

$\frac{- 3 x \ln (17)}{- 3 \ln (17)} = \frac{- \ln (\frac{17}{4})}{- 3 \ln (17)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 3$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 3 x \ln (17)}{- 3 \ln (17)} = \frac{- \ln (\frac{17}{4})}{- 3 \ln (17)}$

Step 9.2.1.2

Rewrite the expression.

$\frac{x \ln (17)}{\ln (17)} = \frac{- \ln (\frac{17}{4})}{- 3 \ln (17)}$

Step 9.2.2

Cancel the common factor of $\ln (17)$ .

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Step 9.2.2.1

Cancel the common factor.

$\frac{x \ln (17)}{\ln (17)} = \frac{- \ln (\frac{17}{4})}{- 3 \ln (17)}$

Step 9.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{17}{4})}{- 3 \ln (17)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Dividing two negative values results in a positive value.

$x = \frac{\ln (\frac{17}{4})}{3 \ln (17)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{17}{4})}{3 \ln (17)}$

Decimal Form:

$x = 0.17023297 \dots$