Calculus Examples

$y = 200 {(1.03)}^{x}$

Step 1

Rewrite the equation as $200 \cdot {1.03}^{x} = y$ .

$200 \cdot {1.03}^{x} = y$

Step 2

Divide each term in $200 \cdot {1.03}^{x} = y$ by $200$ and simplify.

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Step 2.1

Divide each term in $200 \cdot {1.03}^{x} = y$ by $200$ .

$\frac{200 \cdot {1.03}^{x}}{200} = \frac{y}{200}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $200$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{200 \cdot {1.03}^{x}}{200} = \frac{y}{200}$

Step 2.2.1.2

Divide ${1.03}^{x}$ by $1$ .

${1.03}^{x} = \frac{y}{200}$

Step 3

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

$\ln ({1.03}^{x}) = \ln (\frac{y}{200})$

Step 4

Expand $\ln ({1.03}^{x})$ by moving $x$ outside the logarithm.

$x \ln (1.03) = \ln (\frac{y}{200})$

Step 5

Divide each term in $x \ln (1.03) = \ln (\frac{y}{200})$ by $\ln (1.03)$ and simplify.

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Step 5.1

Divide each term in $x \ln (1.03) = \ln (\frac{y}{200})$ by $\ln (1.03)$ .

$\frac{x \ln (1.03)}{\ln (1.03)} = \frac{\ln (\frac{y}{200})}{\ln (1.03)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $\ln (1.03)$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{x \ln (1.03)}{\ln (1.03)} = \frac{\ln (\frac{y}{200})}{\ln (1.03)}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{y}{200})}{\ln (1.03)}$

Step 6

To rewrite as a function of $y$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $y$ is on the other side.

$f (y) = \frac{\ln (\frac{y}{200})}{\ln (1.03)}$