Precalculus Examples

$1000 {(1.005)}^{12 t} \geq 3000$

Step 1

Divide each term in $1000 \cdot {1.005}^{12 t} \geq 3000$ by $1000$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $1000 \cdot {1.005}^{12 t} \geq 3000$ by $1000$ .

$\frac{1000 \cdot {1.005}^{12 t}}{1000} \geq \frac{3000}{1000}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $1000$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{1000 \cdot {1.005}^{12 t}}{1000} \geq \frac{3000}{1000}$

Step 1.2.1.2

Divide ${1.005}^{12 t}$ by $1$ .

${1.005}^{12 t} \geq \frac{3000}{1000}$

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Divide $3000$ by $1000$ .

${1.005}^{12 t} \geq 3$

Step 2

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

$\ln ({1.005}^{12 t}) \geq \ln (3)$

Step 3

Expand $\ln ({1.005}^{12 t})$ by moving $12 t$ outside the logarithm.

$12 t \ln (1.005) \geq \ln (3)$

Step 4

Divide each term in $12 t \ln (1.005) \geq \ln (3)$ by $12 \ln (1.005)$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $12 t \ln (1.005) \geq \ln (3)$ by $12 \ln (1.005)$ .

$\frac{12 t \ln (1.005)}{12 \ln (1.005)} \geq \frac{\ln (3)}{12 \ln (1.005)}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{12 t \ln (1.005)}{12 \ln (1.005)} \geq \frac{\ln (3)}{12 \ln (1.005)}$

Step 4.2.1.2

Rewrite the expression.

$\frac{t \ln (1.005)}{\ln (1.005)} \geq \frac{\ln (3)}{12 \ln (1.005)}$

Step 4.2.2

Cancel the common factor of $\ln (1.005)$ .

Tap for more steps...

Step 4.2.2.1

Cancel the common factor.

$\frac{t \ln (1.005)}{\ln (1.005)} \geq \frac{\ln (3)}{12 \ln (1.005)}$

Step 4.2.2.2

Divide $t$ by $1$ .

$t \geq \frac{\ln (3)}{12 \ln (1.005)}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$t \geq \frac{\ln (3)}{12 \ln (1.005)}$

Interval Notation:

$[\frac{\ln (3)}{12 \ln (1.005)}, \infty)$

Step 6