Algebra Examples

$2 (16^{x + 6}) - 8 = 35$

Step 1

Multiply $2 \cdot 16^{x + 6}$ .

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Step 1.1

Rewrite $16$ as $2^{4}$ .

$2 \cdot {(2^{4})}^{x + 6} - 8 = 35$

Step 1.2

Multiply the exponents in ${(2^{4})}^{x + 6}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 \cdot 2^{4 (x + 6)} - 8 = 35$

Step 1.2.2

Apply the distributive property.

$2 \cdot 2^{4 x + 4 \cdot 6} - 8 = 35$

Step 1.2.3

Multiply $4$ by $6$ .

$2 \cdot 2^{4 x + 24} - 8 = 35$

Step 1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{1 + 4 x + 24} - 8 = 35$

Step 1.4

Add $1$ and $24$ .

$2^{4 x + 25} - 8 = 35$

Step 2

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

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Step 2.1

Add $8$ to both sides of the equation.

$2^{4 x + 25} = 35 + 8$

Step 2.2

Add $35$ and $8$ .

$2^{4 x + 25} = 43$

Step 3

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

$\ln (2^{4 x + 25}) = \ln (43)$

Step 4

Expand $\ln (2^{4 x + 25})$ by moving $4 x + 25$ outside the logarithm.

$(4 x + 25) \ln (2) = \ln (43)$

Step 5

Simplify the left side.

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Step 5.1

Apply the distributive property.

$4 x \ln (2) + 25 \ln (2) = \ln (43)$

Step 6

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

$4 x \ln (2) + 25 \ln (2) - \ln (43) = 0$

Step 7

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

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Step 7.1

Subtract $25 \ln (2)$ from both sides of the equation.

$4 x \ln (2) - \ln (43) = - 25 \ln (2)$

Step 7.2

Add $\ln (43)$ to both sides of the equation.

$4 x \ln (2) = - 25 \ln (2) + \ln (43)$

Step 8

Divide each term in $4 x \ln (2) = - 25 \ln (2) + \ln (43)$ by $4 \ln (2)$ and simplify.

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Step 8.1

Divide each term in $4 x \ln (2) = - 25 \ln (2) + \ln (43)$ by $4 \ln (2)$ .

$\frac{4 x \ln (2)}{4 \ln (2)} = \frac{- 25 \ln (2)}{4 \ln (2)} + \frac{\ln (43)}{4 \ln (2)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $4$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{4 x \ln (2)}{4 \ln (2)} = \frac{- 25 \ln (2)}{4 \ln (2)} + \frac{\ln (43)}{4 \ln (2)}$

Step 8.2.1.2

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 25 \ln (2)}{4 \ln (2)} + \frac{\ln (43)}{4 \ln (2)}$

Step 8.2.2

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

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Step 8.2.2.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 25 \ln (2)}{4 \ln (2)} + \frac{\ln (43)}{4 \ln (2)}$

Step 8.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 25 \ln (2)}{4 \ln (2)} + \frac{\ln (43)}{4 \ln (2)}$

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 $\ln (2)$ .

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Step 8.3.1.1.1

Cancel the common factor.

$x = \frac{- 25 \ln (2)}{4 \ln (2)} + \frac{\ln (43)}{4 \ln (2)}$

Step 8.3.1.1.2

Rewrite the expression.

$x = \frac{- 25}{4} + \frac{\ln (43)}{4 \ln (2)}$

Step 8.3.1.2

Move the negative in front of the fraction.

$x = - \frac{25}{4} + \frac{\ln (43)}{4 \ln (2)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{25}{4} + \frac{\ln (43)}{4 \ln (2)}$

Decimal Form:

$x = - 4.89343381 \dots$