Pre-Algebra Examples

$4000 {(1 + 0.05)}^{t} = 10000$

Step 1

Add $1$ and $0.05$ .

$4000 \cdot {1.05}^{t} = 10000$

Step 2

Divide each term in $4000 \cdot {1.05}^{t} = 10000$ by $4000$ and simplify.

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Step 2.1

Divide each term in $4000 \cdot {1.05}^{t} = 10000$ by $4000$ .

$\frac{4000 \cdot {1.05}^{t}}{4000} = \frac{10000}{4000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4000 \cdot {1.05}^{t}}{4000} = \frac{10000}{4000}$

Step 2.2.1.2

Divide ${1.05}^{t}$ by $1$ .

${1.05}^{t} = \frac{10000}{4000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $10000$ and $4000$ .

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Step 2.3.1.1

Factor $2000$ out of $10000$ .

${1.05}^{t} = \frac{2000 (5)}{4000}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2000$ out of $4000$ .

${1.05}^{t} = \frac{2000 \cdot 5}{2000 \cdot 2}$

Step 2.3.1.2.2

Cancel the common factor.

${1.05}^{t} = \frac{2000 \cdot 5}{2000 \cdot 2}$

Step 2.3.1.2.3

Rewrite the expression.

${1.05}^{t} = \frac{5}{2}$

Step 3

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

$\ln ({1.05}^{t}) = \ln (\frac{5}{2})$

Step 4

Expand $\ln ({1.05}^{t})$ by moving $t$ outside the logarithm.

$t \ln (1.05) = \ln (\frac{5}{2})$

Step 5

Divide each term in $t \ln (1.05) = \ln (\frac{5}{2})$ by $\ln (1.05)$ and simplify.

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Step 5.1

Divide each term in $t \ln (1.05) = \ln (\frac{5}{2})$ by $\ln (1.05)$ .

$\frac{t \ln (1.05)}{\ln (1.05)} = \frac{\ln (\frac{5}{2})}{\ln (1.05)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $\ln (1.05)$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{t \ln (1.05)}{\ln (1.05)} = \frac{\ln (\frac{5}{2})}{\ln (1.05)}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{5}{2})}{\ln (1.05)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\ln (\frac{5}{2})}{\ln (1.05)}$

Decimal Form:

$t = 18.78023465 \dots$