Calculus Examples

$y = \log (2 x - 12)$

Step 1

Rewrite the equation as $\log (2 x - 12) = y$ .

$\log (2 x - 12) = y$

Step 2

Rewrite $\log (2 x - 12) = y$ 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 $b^{y} = x$ .

$10^{y} = 2 x - 12$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $2 x - 12 = 10^{y}$ .

$2 x - 12 = 10^{y}$

Step 3.2

Add $12$ to both sides of the equation.

$2 x = 10^{y} + 12$

Step 3.3

Divide each term in $2 x = 10^{y} + 12$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x = 10^{y} + 12$ by $2$ .

$\frac{2 x}{2} = \frac{10^{y}}{2} + \frac{12}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{10^{y}}{2} + \frac{12}{2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{10^{y}}{2} + \frac{12}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $12$ by $2$ .

$x = \frac{10^{y}}{2} + 6$