Algebra Examples

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

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

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)

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

Step 2

Expand the left side.

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Step 2.1

Expand

ln (e^{x - 8})

$\ln (e^{x - 8})$ by moving

x - 8

$x - 8$ outside the logarithm.

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

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

Step 2.2

The natural logarithm of

e

$e$ is

1

$1$ .

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

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

Step 2.3

Multiply

x - 8

$x - 8$ by

1

$1$ .

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

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

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

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

Step 3

Subtract

ln (sin (x) + y)

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

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

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

Step 4

To solve for

x

$x$ , rewrite the equation using properties of logarithms.

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

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

Step 5

Rewrite

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

$\ln (\sin (x) + y) = x - 8$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b \neq 1

$b \neq 1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

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

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

Step 6

Solve for

x

$x$ .

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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)

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

Step 6.2

Expand the left side.

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Step 6.2.1

Expand

ln (e^{x - 8})

$\ln (e^{x - 8})$ by moving

x - 8

$x - 8$ outside the logarithm.

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

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

Step 6.2.2

The natural logarithm of

e

$e$ is

1

$1$ .

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

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

Step 6.2.3

Multiply

x - 8

$x - 8$ by

1

$1$ .

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

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

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

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

Step 6.3

Subtract

ln (sin (x) + y)

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

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

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

Step 6.4

To solve for

x

$x$ , rewrite the equation using properties of logarithms.

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

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

Step 6.5

Rewrite

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

$\ln (\sin (x) + y) = x - 8$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b \neq 1

$b \neq 1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

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

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

Step 6.6

Solve for

x

$x$ .

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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)

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

Step 6.6.2

Expand the left side.

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Step 6.6.2.1

Expand

ln (e^{x - 8})

$\ln (e^{x - 8})$ by moving

x - 8

$x - 8$ outside the logarithm.

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

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

Step 6.6.2.2

The natural logarithm of

e

$e$ is

1

$1$ .

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

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

Step 6.6.2.3

Multiply

x - 8

$x - 8$ by

1

$1$ .

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

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

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

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

Step 6.6.3

Subtract

ln (sin (x) + y)

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

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

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

Step 6.6.4

To solve for

x

$x$ , rewrite the equation using properties of logarithms.

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

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

Step 6.6.5

Rewrite

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

$\ln (\sin (x) + y) = x - 8$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b \neq 1

$b \neq 1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

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

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

Step 6.6.6

Solve for

x

$x$ .

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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)

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

Step 6.6.6.2

Expand the left side.

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Step 6.6.6.2.1

Expand

ln (e^{x - 8})

$\ln (e^{x - 8})$ by moving

x - 8

$x - 8$ outside the logarithm.

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

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

Step 6.6.6.2.2

The natural logarithm of

e

$e$ is

1

$1$ .

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

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

Step 6.6.6.2.3

Multiply

x - 8

$x - 8$ by

1

$1$ .

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

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

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

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

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

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

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

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

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

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