Precalculus Examples

$8 (3^{6 - x}) = 40$

Step 1

Divide each term in $8 \cdot 3^{6 - x} = 40$ by $8$ and simplify.

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Step 1.1

Divide each term in $8 \cdot 3^{6 - x} = 40$ by $8$ .

$\frac{8 \cdot 3^{6 - x}}{8} = \frac{40}{8}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $8$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{8 \cdot 3^{6 - x}}{8} = \frac{40}{8}$

Step 1.2.1.2

Divide $3^{6 - x}$ by $1$ .

$3^{6 - x} = \frac{40}{8}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $40$ by $8$ .

$3^{6 - x} = 5$

Step 2

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

$\ln (3^{6 - x}) = \ln (5)$

Step 3

Expand $\ln (3^{6 - x})$ by moving $6 - x$ outside the logarithm.

$(6 - x) \ln (3) = \ln (5)$

Step 4

Simplify the left side.

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Step 4.1

Apply the distributive property.

$6 \ln (3) - x \ln (3) = \ln (5)$

Step 5

Reorder $6 \ln (3)$ and $- x \ln (3)$ .

$- x \ln (3) + 6 \ln (3) = \ln (5)$

Step 6

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

$- x \ln (3) + 6 \ln (3) - \ln (5) = 0$

Step 7

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

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Step 7.1

Subtract $6 \ln (3)$ from both sides of the equation.

$- x \ln (3) - \ln (5) = - 6 \ln (3)$

Step 7.2

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

$- x \ln (3) = - 6 \ln (3) + \ln (5)$

Step 8

Divide each term in $- x \ln (3) = - 6 \ln (3) + \ln (5)$ by $- \ln (3)$ and simplify.

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Step 8.1

Divide each term in $- x \ln (3) = - 6 \ln (3) + \ln (5)$ by $- \ln (3)$ .

$\frac{- x \ln (3)}{- \ln (3)} = \frac{- 6 \ln (3)}{- \ln (3)} + \frac{\ln (5)}{- \ln (3)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Dividing two negative values results in a positive value.

$\frac{x \ln (3)}{\ln (3)} = \frac{- 6 \ln (3)}{- \ln (3)} + \frac{\ln (5)}{- \ln (3)}$

Step 8.2.2

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

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Step 8.2.2.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{- 6 \ln (3)}{- \ln (3)} + \frac{\ln (5)}{- \ln (3)}$

Step 8.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 6 \ln (3)}{- \ln (3)} + \frac{\ln (5)}{- \ln (3)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

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

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Step 8.3.1.1.1

Cancel the common factor.

$x = \frac{- 6 \ln (3)}{- \ln (3)} + \frac{\ln (5)}{- \ln (3)}$

Step 8.3.1.1.2

Rewrite the expression.

$x = \frac{- 6}{- 1} + \frac{\ln (5)}{- \ln (3)}$

Step 8.3.1.1.3

Move the negative one from the denominator of $\frac{- 6}{- 1}$ .

$x = - 1 \cdot - 6 + \frac{\ln (5)}{- \ln (3)}$

Step 8.3.1.2

Rewrite $- 1 \cdot - 6$ as $- - 6$ .

$x = - - 6 + \frac{\ln (5)}{- \ln (3)}$

Step 8.3.1.3

Multiply $- 1$ by $- 6$ .

$x = 6 + \frac{\ln (5)}{- \ln (3)}$

Step 8.3.1.4

Move the negative in front of the fraction.

$x = 6 - \frac{\ln (5)}{\ln (3)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 6 - \frac{\ln (5)}{\ln (3)}$

Decimal Form:

$x = 4.53502647 \dots$