Precalculus Examples

$12^{x - 4} > 4^{x}$

Step 1

Take the log of both sides of the inequality.

$\ln (12^{x - 4}) > \ln (4^{x})$

Step 2

Expand $\ln (12^{x - 4})$ by moving $x - 4$ outside the logarithm.

$(x - 4) \ln (12) > \ln (4^{x})$

Step 3

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

$(x - 4) \ln (12) > x \ln (4)$

Step 4

Solve the inequality for $x$ .

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Step 4.1

Simplify $(x - 4) \ln (12)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (x - 4) \ln (12) > x \ln (4)$

Step 4.1.2

Simplify by adding zeros.

$(x - 4) \ln (12) > x \ln (4)$

Step 4.1.3

Apply the distributive property.

$x \ln (12) - 4 \ln (12) > x \ln (4)$

Step 4.2

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

$x \ln (12) - 4 \ln (12) - x \ln (4) > 0$

Step 4.3

Add $4 \ln (12)$ to both sides of the inequality.

$x \ln (12) - x \ln (4) > 4 \ln (12)$

Step 4.4

Factor $x$ out of $x \ln (12) - x \ln (4)$ .

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Step 4.4.1

Factor $x$ out of $x \ln (12)$ .

$x (\ln (12)) - x \ln (4) > 4 \ln (12)$

Step 4.4.2

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

$x (\ln (12)) + x (- 1 \ln (4)) > 4 \ln (12)$

Step 4.4.3

Factor $x$ out of $x (\ln (12)) + x (- 1 \ln (4))$ .

$x (\ln (12) - 1 \ln (4)) > 4 \ln (12)$

Step 4.5

Rewrite $- 1 \ln (4)$ as $- \ln (4)$ .

$x (\ln (12) - \ln (4)) > 4 \ln (12)$

Step 4.6

Divide each term in $x (\ln (12) - \ln (4)) > 4 \ln (12)$ by $\ln (12) - \ln (4)$ and simplify.

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Step 4.6.1

Divide each term in $x (\ln (12) - \ln (4)) > 4 \ln (12)$ by $\ln (12) - \ln (4)$ .

$\frac{x (\ln (12) - \ln (4))}{\ln (12) - \ln (4)} > \frac{4 \ln (12)}{\ln (12) - \ln (4)}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $\ln (12) - \ln (4)$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{x (\ln (12) - \ln (4))}{\ln (12) - \ln (4)} > \frac{4 \ln (12)}{\ln (12) - \ln (4)}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x > \frac{4 \ln (12)}{\ln (12) - \ln (4)}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x > \frac{4 \ln (12)}{\ln (12) - \ln (4)}$

Interval Notation:

$(\frac{4 \ln (12)}{\ln (12) - \ln (4)}, \infty)$

Step 6