Precalculus Examples

$6 (2^{3 x - 1}) - 7 = 9$

Step 1

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

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Step 1.1

Add $7$ to both sides of the equation.

$6 \cdot 2^{3 x - 1} = 9 + 7$

Step 1.2

Add $9$ and $7$ .

$6 \cdot 2^{3 x - 1} = 16$

Step 2

Divide each term in $6 \cdot 2^{3 x - 1} = 16$ by $6$ and simplify.

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Step 2.1

Divide each term in $6 \cdot 2^{3 x - 1} = 16$ by $6$ .

$\frac{6 \cdot 2^{3 x - 1}}{6} = \frac{16}{6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $6$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{6 \cdot 2^{3 x - 1}}{6} = \frac{16}{6}$

Step 2.2.1.2

Divide $2^{3 x - 1}$ by $1$ .

$2^{3 x - 1} = \frac{16}{6}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $16$ and $6$ .

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Step 2.3.1.1

Factor $2$ out of $16$ .

$2^{3 x - 1} = \frac{2 (8)}{6}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $6$ .

$2^{3 x - 1} = \frac{2 \cdot 8}{2 \cdot 3}$

Step 2.3.1.2.2

Cancel the common factor.

$2^{3 x - 1} = \frac{2 \cdot 8}{2 \cdot 3}$

Step 2.3.1.2.3

Rewrite the expression.

$2^{3 x - 1} = \frac{8}{3}$

Step 3

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

$\ln (2^{3 x - 1}) = \ln (\frac{8}{3})$

Step 4

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

$(3 x - 1) \ln (2) = \ln (\frac{8}{3})$

Step 5

Simplify the left side.

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Step 5.1

Simplify $(3 x - 1) \ln (2)$ .

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Step 5.1.1

Apply the distributive property.

$3 x \ln (2) - 1 \ln (2) = \ln (\frac{8}{3})$

Step 5.1.2

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

$3 x \ln (2) - \ln (2) = \ln (\frac{8}{3})$

Step 6

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

$3 x \ln (2) - \ln (2) - \ln (\frac{8}{3}) = 0$

Step 7

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

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Step 7.1

Add $\ln (2)$ to both sides of the equation.

$3 x \ln (2) - \ln (\frac{8}{3}) = \ln (2)$

Step 7.2

Add $\ln (\frac{8}{3})$ to both sides of the equation.

$3 x \ln (2) = \ln (2) + \ln (\frac{8}{3})$

Step 8

Divide each term in $3 x \ln (2) = \ln (2) + \ln (\frac{8}{3})$ by $3 \ln (2)$ and simplify.

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Step 8.1

Divide each term in $3 x \ln (2) = \ln (2) + \ln (\frac{8}{3})$ by $3 \ln (2)$ .

$\frac{3 x \ln (2)}{3 \ln (2)} = \frac{\ln (2)}{3 \ln (2)} + \frac{\ln (\frac{8}{3})}{3 \ln (2)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $3$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{3 x \ln (2)}{3 \ln (2)} = \frac{\ln (2)}{3 \ln (2)} + \frac{\ln (\frac{8}{3})}{3 \ln (2)}$

Step 8.2.1.2

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (2)}{3 \ln (2)} + \frac{\ln (\frac{8}{3})}{3 \ln (2)}$

Step 8.2.2

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

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Step 8.2.2.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (2)}{3 \ln (2)} + \frac{\ln (\frac{8}{3})}{3 \ln (2)}$

Step 8.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (2)}{3 \ln (2)} + \frac{\ln (\frac{8}{3})}{3 \ln (2)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

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

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Step 8.3.1.1

Cancel the common factor.

$x = \frac{\ln (2)}{3 \ln (2)} + \frac{\ln (\frac{8}{3})}{3 \ln (2)}$

Step 8.3.1.2

Rewrite the expression.

$x = \frac{1}{3} + \frac{\ln (\frac{8}{3})}{3 \ln (2)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3} + \frac{\ln (\frac{8}{3})}{3 \ln (2)}$

Decimal Form:

$x = 0.80501249 \dots$