Precalculus Examples

$x = e^{2 t}$ , $y = e^{t}$

Step 1

Solve the equation for $e$ .

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Step 1.1

Rewrite the equation as $e^{2 t} = x$ .

$e^{2 t} = x$

$y = e^{t}$

Step 1.2

Subtract $e^{2 t}$ from both sides of the equation.

$0 = x - e^{2 t}$

$y = e^{t}$

Step 1.3

Rewrite the equation as $x - e^{2 t} = 0$ .

$x - e^{2 t} = 0$

$y = e^{t}$

$x - e^{2 t} = 0$

$y = e^{t}$

Step 2

Remove parentheses.

$y = 0^{t}$

$x - e^{2 t} = 0$

Step 3

Solve the equation for $t$ .

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Step 3.1

Rewrite the equation as $0^{2 t} = x$ .

$0^{2 t} = x$

$x - e^{2 t} = 0$

Step 3.2

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

$\ln (0^{2 t}) = \ln (x)$

$x - e^{2 t} = 0$

Step 3.3

Expand $\ln (0^{2 t})$ by moving $2 t$ outside the logarithm.

$2 t \ln (0) = \ln (x)$

$x - e^{2 t} = 0$

Step 3.4

Simplify the left side.

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Step 3.4.1

The equation cannot be solved because it is undefined.

Undefined

$x - e^{2 t} = 0$

Undefined

$x - e^{2 t} = 0$

Step 3.5

Subtract $Undefined$ from both sides of the equation.

$0 = \ln (x) - Undefined$

$x - e^{2 t} = 0$

Step 3.6

Rewrite the equation as $\ln (x) - Undefined = 0$ .

$\ln (x) - Undefined = 0$

$x - e^{2 t} = 0$

Step 3.7

Expand $\ln (0^{2 t})$ by moving $2 t$ outside the logarithm.

$2 t \ln (0) = \ln (x)$

$x - e^{2 t} = 0$

Step 3.8

The variable $t$ got canceled.

All real numbers

$x - e^{2 t} = 0$

All real numbers

$x - e^{2 t} = 0$