Algebra Examples

$e^{x - 8} = \sin (x) + y$

Step 1

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

$\ln (e^{x - 8}) = \ln (\sin (x) + y)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{x - 8})$ by moving $x - 8$ outside the logarithm.

$(x - 8) \ln (e) = \ln (\sin (x) + y)$

Step 2.2

The natural logarithm of $e$ is $1$ .

$(x - 8) \cdot 1 = \ln (\sin (x) + y)$

Step 2.3

Multiply $x - 8$ by $1$ .

$x - 8 = \ln (\sin (x) + y)$

Step 3

Subtract $\ln (\sin (x) + y)$ from both sides of the equation.

$x - 8 - \ln (\sin (x) + y) = 0$

Step 4

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (\sin (x) + y)} = e^{x - 8}$

Step 5

Rewrite $\ln (\sin (x) + y) = x - 8$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{x - 8} = \sin (x) + y$

Step 6

Solve for $x$ .

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Step 6.1

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

$\ln (e^{x - 8}) = \ln (\sin (x) + y)$

Step 6.2

Expand the left side.

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Step 6.2.1

Expand $\ln (e^{x - 8})$ by moving $x - 8$ outside the logarithm.

$(x - 8) \ln (e) = \ln (\sin (x) + y)$

Step 6.2.2

The natural logarithm of $e$ is $1$ .

$(x - 8) \cdot 1 = \ln (\sin (x) + y)$

Step 6.2.3

Multiply $x - 8$ by $1$ .

$x - 8 = \ln (\sin (x) + y)$

Step 6.3

Subtract $\ln (\sin (x) + y)$ from both sides of the equation.

$x - 8 - \ln (\sin (x) + y) = 0$

Step 6.4

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (\sin (x) + y)} = e^{x - 8}$

Step 6.5

$e^{x - 8} = \sin (x) + y$

Step 6.6

Solve for $x$ .

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Step 6.6.1

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

$\ln (e^{x - 8}) = \ln (\sin (x) + y)$

Step 6.6.2

Expand the left side.

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Step 6.6.2.1

Expand $\ln (e^{x - 8})$ by moving $x - 8$ outside the logarithm.

$(x - 8) \ln (e) = \ln (\sin (x) + y)$

Step 6.6.2.2

The natural logarithm of $e$ is $1$ .

$(x - 8) \cdot 1 = \ln (\sin (x) + y)$

Step 6.6.2.3

Multiply $x - 8$ by $1$ .

$x - 8 = \ln (\sin (x) + y)$

Step 6.6.3

Subtract $\ln (\sin (x) + y)$ from both sides of the equation.

$x - 8 - \ln (\sin (x) + y) = 0$

Step 6.6.4

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (\sin (x) + y)} = e^{x - 8}$

Step 6.6.5

$e^{x - 8} = \sin (x) + y$

Step 6.6.6

Solve for $x$ .

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Step 6.6.6.1

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

$\ln (e^{x - 8}) = \ln (\sin (x) + y)$

Step 6.6.6.2

Expand the left side.

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Step 6.6.6.2.1

Expand $\ln (e^{x - 8})$ by moving $x - 8$ outside the logarithm.

$(x - 8) \ln (e) = \ln (\sin (x) + y)$

Step 6.6.6.2.2

The natural logarithm of $e$ is $1$ .

$(x - 8) \cdot 1 = \ln (\sin (x) + y)$

Step 6.6.6.2.3

Multiply $x - 8$ by $1$ .

$x - 8 = \ln (\sin (x) + y)$