Precalculus Examples

$\ln (y - 6) - \ln (8) = x + \ln (x)$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{y - 6}{8}) = x + \ln (x)$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$\ln (\frac{y - 6}{8}) - \ln (x) = x$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{\frac{y - 6}{8}}{x}) = x$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{y - 6}{8} \cdot \frac{1}{x}) = x$

Step 5

Multiply $\frac{y - 6}{8}$ by $\frac{1}{x}$ .

$\ln (\frac{y - 6}{8 x}) = x$

Step 6

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

$e^{\ln (\frac{y - 6}{8 x})} = e^{x}$

Step 7

Rewrite $\ln (\frac{y - 6}{8 x}) = x$ 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 $b^{y} = x$ .

$e^{x} = \frac{y - 6}{8 x}$

Step 8

Solve for $y$ .

Tap for more steps...

Step 8.1

Rewrite the equation as $\frac{y - 6}{8 x} = e^{x}$ .

$\frac{y - 6}{8 x} = e^{x}$

Step 8.2

Multiply both sides by $8 x$ .

$\frac{y - 6}{8 x} (8 x) = e^{x} (8 x)$

Step 8.3

Simplify.

Tap for more steps...

Step 8.3.1

Simplify the left side.

Tap for more steps...

Step 8.3.1.1

Simplify $\frac{y - 6}{8 x} (8 x)$ .

Tap for more steps...

Step 8.3.1.1.1

Rewrite using the commutative property of multiplication.

$8 \frac{y - 6}{8 x} x = e^{x} (8 x)$

Step 8.3.1.1.2

Cancel the common factor of $8$ .

Tap for more steps...

Step 8.3.1.1.2.1

Factor $8$ out of $8 x$ .

$8 \frac{y - 6}{8 (x)} x = e^{x} (8 x)$

Step 8.3.1.1.2.2

Cancel the common factor.

$8 \frac{y - 6}{8 x} x = e^{x} (8 x)$

Step 8.3.1.1.2.3

Rewrite the expression.

$\frac{y - 6}{x} x = e^{x} (8 x)$

Step 8.3.1.1.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 8.3.1.1.3.1

Cancel the common factor.

$\frac{y - 6}{x} x = e^{x} (8 x)$

Step 8.3.1.1.3.2

Rewrite the expression.

$y - 6 = e^{x} (8 x)$

Step 8.3.2

Simplify the right side.

Tap for more steps...

Step 8.3.2.1

Simplify $e^{x} (8 x)$ .

Tap for more steps...

Step 8.3.2.1.1

Rewrite using the commutative property of multiplication.

$y - 6 = 8 e^{x} x$

Step 8.3.2.1.2

Reorder factors in $8 e^{x} x$ .

$y - 6 = 8 x e^{x}$

Step 8.4

Add $6$ to both sides of the equation.

$y = 8 x e^{x} + 6$