Precalculus Examples

$6^{x} < 3^{x}$

Step 1

Take the log of both sides of the inequality.

$\ln (6^{x}) < \ln (3^{x})$

Step 2

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

$x \ln (6) < \ln (3^{x})$

Step 3

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

$x \ln (6) < x \ln (3)$

Step 4

Solve the inequality for $x$ .

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Step 4.1

Subtract $x \ln (3)$ from both sides of the inequality.

$x \ln (6) - x \ln (3) < 0$

Step 4.2

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

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Step 4.2.1

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

$x (\ln (6)) - x \ln (3) < 0$

Step 4.2.2

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

$x (\ln (6)) + x (- 1 \ln (3)) < 0$

Step 4.2.3

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

$x (\ln (6) - 1 \ln (3)) < 0$

Step 4.3

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

$x (\ln (6) - \ln (3)) < 0$

Step 4.4

Divide each term in $x (\ln (6) - \ln (3)) < 0$ by $\ln (6) - \ln (3)$ and simplify.

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Step 4.4.1

Divide each term in $x (\ln (6) - \ln (3)) < 0$ by $\ln (6) - \ln (3)$ .

$\frac{x (\ln (6) - \ln (3))}{\ln (6) - \ln (3)} < \frac{0}{\ln (6) - \ln (3)}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

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

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{x (\ln (6) - \ln (3))}{\ln (6) - \ln (3)} < \frac{0}{\ln (6) - \ln (3)}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x < \frac{0}{\ln (6) - \ln (3)}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Divide $0$ by $\ln (6) - \ln (3)$ .

$x < 0$

Step 5

Convert the inequality to interval notation.

$(- \infty, 0)$

Step 6