Calculus Examples

$4^{x} - 14 \cdot 2^{x} = - 49$

Step 1

Add $49$ to both sides of the equation.

$4^{x} - 14 \cdot 2^{x} + 49 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Rewrite $4^{x}$ as ${(2^{2})}^{x}$ .

${(2^{2})}^{x} - 14 \cdot 2^{x} + 49 = 0$

Step 2.2

Rewrite ${(2^{2})}^{x}$ as ${(2^{x})}^{2}$ .

${(2^{x})}^{2} - 14 \cdot 2^{x} + 49 = 0$

Step 2.3

Let $u = 2^{x}$ . Substitute $u$ for all occurrences of $2^{x}$ .

$u^{2} - 14 u + 49 = 0$

Step 2.4

Factor using the perfect square rule.

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Step 2.4.1

Rewrite $49$ as $7^{2}$ .

$u^{2} - 14 u + 7^{2} = 0$

Step 2.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 u = 2 \cdot u \cdot 7$

Step 2.4.3

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 7 + 7^{2} = 0$

Step 2.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 7$ .

${(u - 7)}^{2} = 0$

Step 2.5

Replace all occurrences of $u$ with $2^{x}$ .

${(2^{x} - 7)}^{2} = 0$

Step 3

Set the $2^{x} - 7$ equal to $0$ .

$2^{x} - 7 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $7$ to both sides of the equation.

$2^{x} = 7$

Step 4.2

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

$\ln (2^{x}) = \ln (7)$

Step 4.3

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

$x \ln (2) = \ln (7)$

Step 4.4

Divide each term in $x \ln (2) = \ln (7)$ by $\ln (2)$ and simplify.

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Step 4.4.1

Divide each term in $x \ln (2) = \ln (7)$ by $\ln (2)$ .

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (7)}{\ln (2)}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

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

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (7)}{\ln (2)}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (7)}{\ln (2)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (7)}{\ln (2)}$

Decimal Form:

$x = 2.80735492 \dots$