Basic Math Examples

$700 = (4000) {(0.65)}^{z}$

Step 1

Rewrite the equation as $4000 \cdot {0.65}^{z} = 700$ .

$4000 \cdot {0.65}^{z} = 700$

Step 2

Divide each term in $4000 \cdot {0.65}^{z} = 700$ by $4000$ and simplify.

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Step 2.1

Divide each term in $4000 \cdot {0.65}^{z} = 700$ by $4000$ .

$\frac{4000 \cdot {0.65}^{z}}{4000} = \frac{700}{4000}$

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.

$\frac{4000 \cdot {0.65}^{z}}{4000} = \frac{700}{4000}$

Step 2.2.1.2

Divide ${0.65}^{z}$ by $1$ .

${0.65}^{z} = \frac{700}{4000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $700$ and $4000$ .

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Step 2.3.1.1

Factor $100$ out of $700$ .

${0.65}^{z} = \frac{100 (7)}{4000}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $100$ out of $4000$ .

${0.65}^{z} = \frac{100 \cdot 7}{100 \cdot 40}$

Step 2.3.1.2.2

Cancel the common factor.

${0.65}^{z} = \frac{100 \cdot 7}{100 \cdot 40}$

Step 2.3.1.2.3

Rewrite the expression.

${0.65}^{z} = \frac{7}{40}$

Step 3

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

$\ln ({0.65}^{z}) = \ln (\frac{7}{40})$

Step 4

Expand $\ln ({0.65}^{z})$ by moving $z$ outside the logarithm.

$z \ln (0.65) = \ln (\frac{7}{40})$

Step 5

Divide each term in $z \ln (0.65) = \ln (\frac{7}{40})$ by $\ln (0.65)$ and simplify.

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Step 5.1

Divide each term in $z \ln (0.65) = \ln (\frac{7}{40})$ by $\ln (0.65)$ .

$\frac{z \ln (0.65)}{\ln (0.65)} = \frac{\ln (\frac{7}{40})}{\ln (0.65)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $\ln (0.65)$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{z \ln (0.65)}{\ln (0.65)} = \frac{\ln (\frac{7}{40})}{\ln (0.65)}$

Step 5.2.1.2

Divide $z$ by $1$ .

$z = \frac{\ln (\frac{7}{40})}{\ln (0.65)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$z = \frac{\ln (\frac{7}{40})}{\ln (0.65)}$

Decimal Form:

$z = 4.04605020 \dots$