Algebra Examples

${(0.2)}^{x + 1} < 0.04$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({0.2}^{x + 1}) < \ln (0.04)$

Step 2

Expand $\ln ({0.2}^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (0.2) < \ln (0.04)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(x + 1) \ln (0.2)$ .

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Step 3.1.1

Apply the distributive property.

$x \ln (0.2) + 1 \ln (0.2) < \ln (0.04)$

Step 3.1.2

Multiply $\ln (0.2)$ by $1$ .

$x \ln (0.2) + \ln (0.2) < \ln (0.04)$

Step 4

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (0.2) + \ln (0.2) - \ln (0.04) = 0$

Step 5

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (0.2) + \ln (\frac{0.2}{0.04}) = 0$

Step 6

Divide $0.2$ by $0.04$ .

$x \ln (0.2) + \ln (5) = 0$

Step 7

Subtract $\ln (5)$ from both sides of the equation.

$x \ln (0.2) = - \ln (5)$

Step 8

Divide each term in $x \ln (0.2) = - \ln (5)$ by $\ln (0.2)$ and simplify.

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Step 8.1

Divide each term in $x \ln (0.2) = - \ln (5)$ by $\ln (0.2)$ .

$\frac{x \ln (0.2)}{\ln (0.2)} = \frac{- \ln (5)}{\ln (0.2)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $\ln (0.2)$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{x \ln (0.2)}{\ln (0.2)} = \frac{- \ln (5)}{\ln (0.2)}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (5)}{\ln (0.2)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (5)}{\ln (0.2)}$

Step 9

The solution consists of all of the true intervals.

$x > - \frac{\ln (5)}{\ln (0.2)}$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x > - \frac{\ln (5)}{\ln (0.2)}$

Interval Notation:

$(- \frac{\ln (5)}{\ln (0.2)}, \infty)$

Step 11