Algebra Examples

${(\frac{1}{3})}^{x} < {(\frac{1}{5})}^{x}$

Step 1

Take the log of both sides of the inequality.

$\ln ({(\frac{1}{3})}^{x}) < \ln ({(\frac{1}{5})}^{x})$

Step 2

Expand $\ln ({(\frac{1}{3})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{1}{3}) < \ln ({(\frac{1}{5})}^{x})$

Step 3

Rewrite $\ln (\frac{1}{3})$ as $\ln (1) - \ln (3)$ .

$x (\ln (1) - \ln (3)) < \ln ({(\frac{1}{5})}^{x})$

Step 4

The natural logarithm of $1$ is $0$ .

$x (0 - \ln (3)) < \ln ({(\frac{1}{5})}^{x})$

Step 5

Subtract $\ln (3)$ from $0$ .

$x (- \ln (3)) < \ln ({(\frac{1}{5})}^{x})$

Step 6

Expand $\ln ({(\frac{1}{5})}^{x})$ by moving $x$ outside the logarithm.

$x (- \ln (3)) < x \ln (\frac{1}{5})$

Step 7

Rewrite $\ln (\frac{1}{5})$ as $\ln (1) - \ln (5)$ .

$x (- \ln (3)) < x (\ln (1) - \ln (5))$

Step 8

The natural logarithm of $1$ is $0$ .

$x (- \ln (3)) < x (0 - \ln (5))$

Step 9

Subtract $\ln (5)$ from $0$ .

$x (- \ln (3)) < x (- \ln (5))$

Step 10

Solve the inequality for $x$ .

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Step 10.1

Rewrite using the commutative property of multiplication.

$- x \ln (3) < x (- \ln (5))$

Step 10.2

Rewrite using the commutative property of multiplication.

$- x \ln (3) < - x \ln (5)$

Step 10.3

Add $x \ln (5)$ to both sides of the inequality.

$- x \ln (3) + x \ln (5) < 0$

Step 10.4

Factor $x$ out of $- x \ln (3) + x \ln (5)$ .

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Step 10.4.1

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

$x (- 1 \ln (3)) + x \ln (5) < 0$

Step 10.4.2

Factor $x$ out of $x \ln (5)$ .

$x (- 1 \ln (3)) + x (\ln (5)) < 0$

Step 10.4.3

Factor $x$ out of $x (- 1 \ln (3)) + x (\ln (5))$ .

$x (- 1 \ln (3) + \ln (5)) < 0$

Step 10.5

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

$x (- \ln (3) + \ln (5)) < 0$

Step 10.6

Divide each term in $x (- \ln (3) + \ln (5)) < 0$ by $- \ln (3) + \ln (5)$ and simplify.

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Step 10.6.1

Divide each term in $x (- \ln (3) + \ln (5)) < 0$ by $- \ln (3) + \ln (5)$ .

$\frac{x (- \ln (3) + \ln (5))}{- \ln (3) + \ln (5)} < \frac{0}{- \ln (3) + \ln (5)}$

Step 10.6.2

Simplify the left side.

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Step 10.6.2.1

Cancel the common factor of $- \ln (3) + \ln (5)$ .

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Step 10.6.2.1.1

Cancel the common factor.

$\frac{x (- \ln (3) + \ln (5))}{- \ln (3) + \ln (5)} < \frac{0}{- \ln (3) + \ln (5)}$

Step 10.6.2.1.2

Divide $x$ by $1$ .

$x < \frac{0}{- \ln (3) + \ln (5)}$

Step 10.6.3

Simplify the right side.

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Step 10.6.3.1

Divide $0$ by $- \ln (3) + \ln (5)$ .

$x < 0$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < 0$

Interval Notation:

$(- \infty, 0)$

Step 12