Finite Math Examples

${(0.28)}^{n} < 0.0001$

Step 1

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

$\ln ({0.28}^{n}) < \ln (0.0001)$

Step 2

Expand $\ln ({0.28}^{n})$ by moving $n$ outside the logarithm.

$n \ln (0.28) < \ln (0.0001)$

Step 3

Divide each term in $n \ln (0.28) < \ln (0.0001)$ by $\ln (0.28)$ and simplify.

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Step 3.1

Divide each term in $n \ln (0.28) < \ln (0.0001)$ by $\ln (0.28)$ .

$\frac{n \ln (0.28)}{\ln (0.28)} < \frac{\ln (0.0001)}{\ln (0.28)}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $\ln (0.28)$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{n \ln (0.28)}{\ln (0.28)} < \frac{\ln (0.0001)}{\ln (0.28)}$

Step 3.2.1.2

Divide $n$ by $1$ .

$n < \frac{\ln (0.0001)}{\ln (0.28)}$

Step 4

The solution consists of all of the true intervals.

$n > \frac{\ln (0.0001)}{\ln (0.28)}$

Step 5

Convert the inequality to interval notation.

$(\frac{\ln (0.0001)}{\ln (0.28)}, \infty)$

Step 6