Algebra Examples

$3^{6 x} > 8^{x - 4}$

Step 1

Take the log of both sides of the inequality.

$\ln (3^{6 x}) > \ln (8^{x - 4})$

Step 2

Expand $\ln (3^{6 x})$ by moving $6 x$ outside the logarithm.

$6 x \ln (3) > \ln (8^{x - 4})$

Step 3

Expand $\ln (8^{x - 4})$ by moving $x - 4$ outside the logarithm.

$6 x \ln (3) > (x - 4) \ln (8)$

Step 4

Solve the inequality for $x$ .

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Step 4.1

Apply the distributive property.

$6 x \ln (3) > x \ln (8) - 4 \ln (8)$

Step 4.2

Subtract $x \ln (8)$ from both sides of the inequality.

$6 x \ln (3) - x \ln (8) > - 4 \ln (8)$

Step 4.3

Factor $x$ out of $6 x \ln (3) - x \ln (8)$ .

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Step 4.3.1

Factor $x$ out of $6 x \ln (3)$ .

$x (6 \ln (3)) - x \ln (8) > - 4 \ln (8)$

Step 4.3.2

Factor $x$ out of $- x \ln (8)$ .

$x (6 \ln (3)) + x (- 1 \ln (8)) > - 4 \ln (8)$

Step 4.3.3

Factor $x$ out of $x (6 \ln (3)) + x (- 1 \ln (8))$ .

$x (6 \ln (3) - 1 \ln (8)) > - 4 \ln (8)$

Step 4.4

Rewrite $- 1 \ln (8)$ as $- \ln (8)$ .

$x (6 \ln (3) - \ln (8)) > - 4 \ln (8)$

Step 4.5

Divide each term in $x (6 \ln (3) - \ln (8)) > - 4 \ln (8)$ by $6 \ln (3) - \ln (8)$ and simplify.

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Step 4.5.1

Divide each term in $x (6 \ln (3) - \ln (8)) > - 4 \ln (8)$ by $6 \ln (3) - \ln (8)$ .

$\frac{x (6 \ln (3) - \ln (8))}{6 \ln (3) - \ln (8)} > \frac{- 4 \ln (8)}{6 \ln (3) - \ln (8)}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $6 \ln (3) - \ln (8)$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{x (6 \ln (3) - \ln (8))}{6 \ln (3) - \ln (8)} > \frac{- 4 \ln (8)}{6 \ln (3) - \ln (8)}$

Step 4.5.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 4 \ln (8)}{6 \ln (3) - \ln (8)}$

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.

$x > - \frac{4 \ln (8)}{6 \ln (3) - \ln (8)}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x > - \frac{4 \ln (8)}{6 \ln (3) - \ln (8)}$

Interval Notation:

$(- \frac{4 \ln (8)}{6 \ln (3) - \ln (8)}, \infty)$

Step 6