Calculus Examples

$11^{x + 2} = 4^{7 x}$

Step 1

Take the log of both sides of the equation.

$\ln (11^{x + 2}) = \ln (4^{7 x})$

Step 2

Expand $\ln (11^{x + 2})$ by moving $x + 2$ outside the logarithm.

$(x + 2) \ln (11) = \ln (4^{7 x})$

Step 3

Expand $\ln (4^{7 x})$ by moving $7 x$ outside the logarithm.

$(x + 2) \ln (11) = 7 x \ln (4)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x + 2) \ln (11)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (x + 2) \ln (11) = 7 x \ln (4)$

Step 4.1.2

Simplify by adding zeros.

$(x + 2) \ln (11) = 7 x \ln (4)$

Step 4.1.3

Apply the distributive property.

$x \ln (11) + 2 \ln (11) = 7 x \ln (4)$

Step 4.2

Subtract $7 x \ln (4)$ from both sides of the equation.

$x \ln (11) + 2 \ln (11) - 7 x \ln (4) = 0$

Step 4.3

Subtract $2 \ln (11)$ from both sides of the equation.

$x \ln (11) - 7 x \ln (4) = - 2 \ln (11)$

Step 4.4

Factor $x$ out of $x \ln (11) - 7 x \ln (4)$ .

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Step 4.4.1

Factor $x$ out of $x \ln (11)$ .

$x (\ln (11)) - 7 x \ln (4) = - 2 \ln (11)$

Step 4.4.2

Factor $x$ out of $- 7 x \ln (4)$ .

$x (\ln (11)) + x (- 7 \ln (4)) = - 2 \ln (11)$

Step 4.4.3

Factor $x$ out of $x (\ln (11)) + x (- 7 \ln (4))$ .

$x (\ln (11) - 7 \ln (4)) = - 2 \ln (11)$

Step 4.5

Divide each term in $x (\ln (11) - 7 \ln (4)) = - 2 \ln (11)$ by $\ln (11) - 7 \ln (4)$ and simplify.

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Step 4.5.1

Divide each term in $x (\ln (11) - 7 \ln (4)) = - 2 \ln (11)$ by $\ln (11) - 7 \ln (4)$ .

$\frac{x (\ln (11) - 7 \ln (4))}{\ln (11) - 7 \ln (4)} = \frac{- 2 \ln (11)}{\ln (11) - 7 \ln (4)}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $\ln (11) - 7 \ln (4)$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{x (\ln (11) - 7 \ln (4))}{\ln (11) - 7 \ln (4)} = \frac{- 2 \ln (11)}{\ln (11) - 7 \ln (4)}$

Step 4.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 \ln (11)}{\ln (11) - 7 \ln (4)}$

Step 4.5.3

Simplify the right side.

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Step 4.5.3.1

Move the negative in front of the fraction.

$x = - \frac{2 \ln (11)}{\ln (11) - 7 \ln (4)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2 \ln (11)}{\ln (11) - 7 \ln (4)}$

Decimal Form:

$x = 0.65640324 \dots$