Precalculus Examples

$f (x) = e^{8 x - 7} - 5^{x - 5}$

Step 1

Set $e^{8 x - 7} - 5^{x - 5}$ equal to $0$ .

$e^{8 x - 7} - 5^{x - 5} = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Move $- 5^{x - 5}$ to the right side of the equation by adding it to both sides.

$e^{8 x - 7} = 5^{x - 5}$

Step 2.2

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

$\ln (e^{8 x - 7}) = \ln (5^{x - 5})$

Step 2.3

Expand the left side.

Tap for more steps...

Step 2.3.1

Expand $\ln (e^{8 x - 7})$ by moving $8 x - 7$ outside the logarithm.

$(8 x - 7) \ln (e) = \ln (5^{x - 5})$

Step 2.3.2

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

$(8 x - 7) \cdot 1 = \ln (5^{x - 5})$

Step 2.3.3

Multiply $8 x - 7$ by $1$ .

$8 x - 7 = \ln (5^{x - 5})$

Step 2.4

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

$8 x - 7 = (x - 5) \ln (5)$

Step 2.5

Simplify the right side.

Tap for more steps...

Step 2.5.1

Apply the distributive property.

$8 x - 7 = x \ln (5) - 5 \ln (5)$

Step 2.6

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

$- x \ln (5) + 5 \ln (5) = - 8 x + 7$

Step 2.7

Add $8 x$ to both sides of the equation.

$- x \ln (5) + 5 \ln (5) + 8 x = 7$

Step 2.8

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

$- x \ln (5) + 8 x = 7 - 5 \ln (5)$

Step 2.9

Factor $x$ out of $- x \ln (5) + 8 x$ .

Tap for more steps...

Step 2.9.1

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

$x (- 1 \ln (5)) + 8 x = 7 - 5 \ln (5)$

Step 2.9.2

Factor $x$ out of $8 x$ .

$x (- 1 \ln (5)) + x \cdot 8 = 7 - 5 \ln (5)$

Step 2.9.3

Factor $x$ out of $x (- 1 \ln (5)) + x \cdot 8$ .

$x (- 1 \ln (5) + 8) = 7 - 5 \ln (5)$

Step 2.10

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

$x (- \ln (5) + 8) = 7 - 5 \ln (5)$

Step 2.11

Divide each term in $x (- \ln (5) + 8) = 7 - 5 \ln (5)$ by $- \ln (5) + 8$ and simplify.

Tap for more steps...

Step 2.11.1

Divide each term in $x (- \ln (5) + 8) = 7 - 5 \ln (5)$ by $- \ln (5) + 8$ .

$\frac{x (- \ln (5) + 8)}{- \ln (5) + 8} = \frac{7}{- \ln (5) + 8} + \frac{- 5 \ln (5)}{- \ln (5) + 8}$

Step 2.11.2

Simplify the left side.

Tap for more steps...

Step 2.11.2.1

Cancel the common factor of $- \ln (5) + 8$ .

Tap for more steps...

Step 2.11.2.1.1

Cancel the common factor.

$\frac{x (- \ln (5) + 8)}{- \ln (5) + 8} = \frac{7}{- \ln (5) + 8} + \frac{- 5 \ln (5)}{- \ln (5) + 8}$

Step 2.11.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{- \ln (5) + 8} + \frac{- 5 \ln (5)}{- \ln (5) + 8}$

Step 2.11.3

Simplify the right side.

Tap for more steps...

Step 2.11.3.1

Combine the numerators over the common denominator.

$x = \frac{7 - 5 \ln (5)}{- \ln (5) + 8}$

Step 2.11.3.2

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

$x = \frac{7 - 5 \ln (5)}{- \ln (5) + 8}$

Step 2.11.3.3

Rewrite $8$ as $- 1 (- 8)$ .

$x = \frac{7 - 5 \ln (5)}{- \ln (5) - 1 (- 8)}$

Step 2.11.3.4

Factor $- 1$ out of $- \ln (5) - 1 (- 8)$ .

$x = \frac{7 - 5 \ln (5)}{- (\ln (5) - 8)}$

Step 2.11.3.5

Rewrite negatives.

Tap for more steps...

Step 2.11.3.5.1

Rewrite $- (\ln (5) - 8)$ as $- 1 (\ln (5) - 8)$ .

$x = \frac{7 - 5 \ln (5)}{- 1 (\ln (5) - 8)}$

Step 2.11.3.5.2

Move the negative in front of the fraction.

$x = - \frac{7 - 5 \ln (5)}{\ln (5) - 8}$

$x = - \frac{7 - 5 \ln (5)}{\ln (5) - 8}$ (Multiplicity of $1$ )

Step 3