Precalculus Examples

$f (x) = e^{9 x - 5} - 5^{x - 9}$

Step 1

Set $e^{9 x - 5} - 5^{x - 9}$ equal to $0$ .

$e^{9 x - 5} - 5^{x - 9} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Move $- 5^{x - 9}$ to the right side of the equation by adding it to both sides.

$e^{9 x - 5} = 5^{x - 9}$

Step 2.2

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

$\ln (e^{9 x - 5}) = \ln (5^{x - 9})$

Step 2.3

Expand the left side.

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Step 2.3.1

Expand $\ln (e^{9 x - 5})$ by moving $9 x - 5$ outside the logarithm.

$(9 x - 5) \ln (e) = \ln (5^{x - 9})$

Step 2.3.2

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

$(9 x - 5) \cdot 1 = \ln (5^{x - 9})$

Step 2.3.3

Multiply $9 x - 5$ by $1$ .

$9 x - 5 = \ln (5^{x - 9})$

Step 2.4

Expand $\ln (5^{x - 9})$ by moving $x - 9$ outside the logarithm.

$9 x - 5 = (x - 9) \ln (5)$

Step 2.5

Simplify the right side.

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Step 2.5.1

Apply the distributive property.

$9 x - 5 = x \ln (5) - 9 \ln (5)$

Step 2.6

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

$- x \ln (5) + 9 \ln (5) = - 9 x + 5$

Step 2.7

Add $9 x$ to both sides of the equation.

$- x \ln (5) + 9 \ln (5) + 9 x = 5$

Step 2.8

Subtract $9 \ln (5)$ from both sides of the equation.

$- x \ln (5) + 9 x = 5 - 9 \ln (5)$

Step 2.9

Factor $x$ out of $- x \ln (5) + 9 x$ .

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Step 2.9.1

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

$x (- 1 \ln (5)) + 9 x = 5 - 9 \ln (5)$

Step 2.9.2

Factor $x$ out of $9 x$ .

$x (- 1 \ln (5)) + x \cdot 9 = 5 - 9 \ln (5)$

Step 2.9.3

Factor $x$ out of $x (- 1 \ln (5)) + x \cdot 9$ .

$x (- 1 \ln (5) + 9) = 5 - 9 \ln (5)$

Step 2.10

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

$x (- \ln (5) + 9) = 5 - 9 \ln (5)$

Step 2.11

Divide each term in $x (- \ln (5) + 9) = 5 - 9 \ln (5)$ by $- \ln (5) + 9$ and simplify.

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Step 2.11.1

Divide each term in $x (- \ln (5) + 9) = 5 - 9 \ln (5)$ by $- \ln (5) + 9$ .

$\frac{x (- \ln (5) + 9)}{- \ln (5) + 9} = \frac{5}{- \ln (5) + 9} + \frac{- 9 \ln (5)}{- \ln (5) + 9}$

Step 2.11.2

Simplify the left side.

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Step 2.11.2.1

Cancel the common factor of $- \ln (5) + 9$ .

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Step 2.11.2.1.1

Cancel the common factor.

$\frac{x (- \ln (5) + 9)}{- \ln (5) + 9} = \frac{5}{- \ln (5) + 9} + \frac{- 9 \ln (5)}{- \ln (5) + 9}$

Step 2.11.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{- \ln (5) + 9} + \frac{- 9 \ln (5)}{- \ln (5) + 9}$

Step 2.11.3

Simplify the right side.

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Step 2.11.3.1

Combine the numerators over the common denominator.

$x = \frac{5 - 9 \ln (5)}{- \ln (5) + 9}$

Step 2.11.3.2

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

$x = \frac{5 - 9 \ln (5)}{- \ln (5) + 9}$

Step 2.11.3.3

Rewrite $9$ as $- 1 (- 9)$ .

$x = \frac{5 - 9 \ln (5)}{- \ln (5) - 1 (- 9)}$

Step 2.11.3.4

Factor $- 1$ out of $- \ln (5) - 1 (- 9)$ .

$x = \frac{5 - 9 \ln (5)}{- (\ln (5) - 9)}$

Step 2.11.3.5

Rewrite negatives.

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Step 2.11.3.5.1

Rewrite $- (\ln (5) - 9)$ as $- 1 (\ln (5) - 9)$ .

$x = \frac{5 - 9 \ln (5)}{- 1 (\ln (5) - 9)}$

Step 2.11.3.5.2

Move the negative in front of the fraction.

$x = - \frac{5 - 9 \ln (5)}{\ln (5) - 9}$

Step 3