Calculus Examples

$4 \cdot 8^{x} = 3.5 e^{1 x}$

Step 1

Multiply $4 \cdot 8^{x}$ .

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Step 1.1

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

$2^{2} \cdot 8^{x} = 3.5 e^{1 x}$

Step 1.2

Rewrite $8$ as $2^{3}$ .

$2^{2} \cdot {(2^{3})}^{x} = 3.5 e^{1 x}$

Step 1.3

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

$2^{2} \cdot 2^{3 x} = 3.5 e^{1 x}$

Step 1.4

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

$2^{2 + 3 x} = 3.5 e^{1 x}$

Step 2

Multiply $x$ by $1$ .

$2^{2 + 3 x} = 3.5 e^{x}$

Step 3

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

$\ln (2^{2 + 3 x}) = \ln (3.5 e^{x})$

Step 4

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

$(2 + 3 x) \ln (2) = \ln (3.5 e^{x})$

Step 5

Expand the right side.

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Step 5.1

Rewrite $\ln (3.5 e^{x})$ as $\ln (3.5) + \ln (e^{x})$ .

$(2 + 3 x) \ln (2) = \ln (3.5) + \ln (e^{x})$

Step 5.2

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

$(2 + 3 x) \ln (2) = \ln (3.5) + x \ln (e)$

Step 5.3

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

$(2 + 3 x) \ln (2) = \ln (3.5) + x \cdot 1$

Step 5.4

Multiply $x$ by $1$ .

$(2 + 3 x) \ln (2) = \ln (3.5) + x$

Step 6

Simplify the left side.

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Step 6.1

Apply the distributive property.

$2 \ln (2) + 3 x \ln (2) = \ln (3.5) + x$

Step 7

Reorder $2 \ln (2)$ and $3 x \ln (2)$ .

$3 x \ln (2) + 2 \ln (2) = \ln (3.5) + x$

Step 8

Reorder $\ln (3.5)$ and $x$ .

$3 x \ln (2) + 2 \ln (2) = x + \ln (3.5)$

Step 9

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

$3 x \ln (2) + 2 \ln (2) - \ln (3.5) = x$

Step 10

Subtract $x$ from both sides of the equation.

$3 x \ln (2) + 2 \ln (2) - \ln (3.5) - x = 0$

Step 11

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

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Step 11.1

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

$3 x \ln (2) - \ln (3.5) - x = - 2 \ln (2)$

Step 11.2

Add $\ln (3.5)$ to both sides of the equation.

$3 x \ln (2) - x = - 2 \ln (2) + \ln (3.5)$

Step 12

Factor $x$ out of $3 x \ln (2) - x$ .

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Step 12.1

Factor $x$ out of $3 x \ln (2)$ .

$x (3 \ln (2)) - x = - 2 \ln (2) + \ln (3.5)$

Step 12.2

Factor $x$ out of $- x$ .

$x (3 \ln (2)) + x \cdot - 1 = - 2 \ln (2) + \ln (3.5)$

Step 12.3

Factor $x$ out of $x (3 \ln (2)) + x \cdot - 1$ .

$x (3 \ln (2) - 1) = - 2 \ln (2) + \ln (3.5)$

Step 13

Divide each term in $x (3 \ln (2) - 1) = - 2 \ln (2) + \ln (3.5)$ by $3 \ln (2) - 1$ and simplify.

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Step 13.1

Divide each term in $x (3 \ln (2) - 1) = - 2 \ln (2) + \ln (3.5)$ by $3 \ln (2) - 1$ .

$\frac{x (3 \ln (2) - 1)}{3 \ln (2) - 1} = \frac{- 2 \ln (2)}{3 \ln (2) - 1} + \frac{\ln (3.5)}{3 \ln (2) - 1}$

Step 13.2

Simplify the left side.

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Step 13.2.1

Cancel the common factor of $3 \ln (2) - 1$ .

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Step 13.2.1.1

Cancel the common factor.

$\frac{x (3 \ln (2) - 1)}{3 \ln (2) - 1} = \frac{- 2 \ln (2)}{3 \ln (2) - 1} + \frac{\ln (3.5)}{3 \ln (2) - 1}$

Step 13.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 \ln (2)}{3 \ln (2) - 1} + \frac{\ln (3.5)}{3 \ln (2) - 1}$

Step 13.3

Simplify the right side.

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Step 13.3.1

Move the negative in front of the fraction.

$x = - \frac{2 \ln (2)}{3 \ln (2) - 1} + \frac{\ln (3.5)}{3 \ln (2) - 1}$

Step 13.3.2

Combine the numerators over the common denominator.

$x = \frac{- 2 \ln (2) + \ln (3.5)}{3 \ln (2) - 1}$

Step 13.3.3

Factor $- 1$ out of $- 2 \ln (2)$ .

$x = \frac{- (2 \ln (2)) + \ln (3.5)}{3 \ln (2) - 1}$

Step 13.3.4

Factor $- 1$ out of $\ln (3.5)$ .

$x = \frac{- (2 \ln (2)) - 1 (- \ln (3.5))}{3 \ln (2) - 1}$

Step 13.3.5

Factor $- 1$ out of $- (2 \ln (2)) - 1 (- \ln (3.5))$ .

$x = \frac{- (2 \ln (2) - \ln (3.5))}{3 \ln (2) - 1}$

Step 13.3.6

Simplify the expression.

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Step 13.3.6.1

Rewrite $- (2 \ln (2) - \ln (3.5))$ as $- 1 (2 \ln (2) - \ln (3.5))$ .

$x = \frac{- 1 (2 \ln (2) - \ln (3.5))}{3 \ln (2) - 1}$

Step 13.3.6.2

Move the negative in front of the fraction.

$x = - \frac{2 \ln (2) - \ln (3.5)}{3 \ln (2) - 1}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2 \ln (2) - \ln (3.5)}{3 \ln (2) - 1}$

Decimal Form:

$x = - 0.12370414 \dots$