Precalculus Examples

$3^{x + 6} = 5^{x - 1}$

Step 1

Take the log of both sides of the equation.

$\ln (3^{x + 6}) = \ln (5^{x - 1})$

Step 2

Expand $\ln (3^{x + 6})$ by moving $x + 6$ outside the logarithm.

$(x + 6) \ln (3) = \ln (5^{x - 1})$

Step 3

Expand $\ln (5^{x - 1})$ by moving $x - 1$ outside the logarithm.

$(x + 6) \ln (3) = (x - 1) \ln (5)$

Step 4

Solve the equation for $x$ .

Tap for more steps...

Step 4.1

Simplify $(x + 6) \ln (3)$ .

Tap for more steps...

Step 4.1.1

Rewrite.

$0 + 0 + (x + 6) \ln (3) = (x - 1) \ln (5)$

Step 4.1.2

Simplify by adding zeros.

$(x + 6) \ln (3) = (x - 1) \ln (5)$

Step 4.1.3

Apply the distributive property.

$x \ln (3) + 6 \ln (3) = (x - 1) \ln (5)$

Step 4.2

Simplify $(x - 1) \ln (5)$ .

Tap for more steps...

Step 4.2.1

Apply the distributive property.

$x \ln (3) + 6 \ln (3) = x \ln (5) - 1 \ln (5)$

Step 4.2.2

Rewrite $- 1 \ln (5)$ as $- \ln (5)$ .

$x \ln (3) + 6 \ln (3) = x \ln (5) - \ln (5)$

Step 4.3

Subtract $x \ln (5)$ from both sides of the equation.

$x \ln (3) + 6 \ln (3) - x \ln (5) = - \ln (5)$

Step 4.4

Subtract $6 \ln (3)$ from both sides of the equation.

$x \ln (3) - x \ln (5) = - \ln (5) - 6 \ln (3)$

Step 4.5

Factor $x$ out of $x \ln (3) - x \ln (5)$ .

Tap for more steps...

Step 4.5.1

Factor $x$ out of $x \ln (3)$ .

$x (\ln (3)) - x \ln (5) = - \ln (5) - 6 \ln (3)$

Step 4.5.2

Factor $x$ out of $- x \ln (5)$ .

$x (\ln (3)) + x (- 1 \ln (5)) = - \ln (5) - 6 \ln (3)$

Step 4.5.3

Factor $x$ out of $x (\ln (3)) + x (- 1 \ln (5))$ .

$x (\ln (3) - 1 \ln (5)) = - \ln (5) - 6 \ln (3)$

Step 4.6

Rewrite $- 1 \ln (5)$ as $- \ln (5)$ .

$x (\ln (3) - \ln (5)) = - \ln (5) - 6 \ln (3)$

Step 4.7

Divide each term in $x (\ln (3) - \ln (5)) = - \ln (5) - 6 \ln (3)$ by $\ln (3) - \ln (5)$ and simplify.

Tap for more steps...

Step 4.7.1

Divide each term in $x (\ln (3) - \ln (5)) = - \ln (5) - 6 \ln (3)$ by $\ln (3) - \ln (5)$ .

$\frac{x (\ln (3) - \ln (5))}{\ln (3) - \ln (5)} = \frac{- \ln (5)}{\ln (3) - \ln (5)} + \frac{- 6 \ln (3)}{\ln (3) - \ln (5)}$

Step 4.7.2

Simplify the left side.

Tap for more steps...

Step 4.7.2.1

Cancel the common factor of $\ln (3) - \ln (5)$ .

Tap for more steps...

Step 4.7.2.1.1

Cancel the common factor.

$\frac{x (\ln (3) - \ln (5))}{\ln (3) - \ln (5)} = \frac{- \ln (5)}{\ln (3) - \ln (5)} + \frac{- 6 \ln (3)}{\ln (3) - \ln (5)}$

Step 4.7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (5)}{\ln (3) - \ln (5)} + \frac{- 6 \ln (3)}{\ln (3) - \ln (5)}$

Step 4.7.3

Simplify the right side.

Tap for more steps...

Step 4.7.3.1

Combine the numerators over the common denominator.

$x = \frac{- \ln (5) - 6 \ln (3)}{\ln (3) - \ln (5)}$

Step 4.7.3.2

Factor $- 1$ out of $- \ln (5)$ .

$x = \frac{- \ln (5) - 6 \ln (3)}{\ln (3) - \ln (5)}$

Step 4.7.3.3

Factor $- 1$ out of $- 6 \ln (3)$ .

$x = \frac{- \ln (5) - (6 \ln (3))}{\ln (3) - \ln (5)}$

Step 4.7.3.4

Factor $- 1$ out of $- \ln (5) - (6 \ln (3))$ .

$x = \frac{- (\ln (5) + 6 \ln (3))}{\ln (3) - \ln (5)}$

Step 4.7.3.5

Simplify the expression.

Tap for more steps...

Step 4.7.3.5.1

Rewrite $- (\ln (5) + 6 \ln (3))$ as $- 1 (\ln (5) + 6 \ln (3))$ .

$x = \frac{- 1 (\ln (5) + 6 \ln (3))}{\ln (3) - \ln (5)}$

Step 4.7.3.5.2

Move the negative in front of the fraction.

$x = - \frac{\ln (5) + 6 \ln (3)}{\ln (3) - \ln (5)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (5) + 6 \ln (3)}{\ln (3) - \ln (5)}$

Decimal Form:

$x = 16.05462072 \dots$