Precalculus Examples

5^{x - 2} = 3^{3 x + 2}

$5^{x - 2} = 3^{3 x + 2}$

Step 1

Take the log of both sides of the equation.

ln (5^{x - 2}) = ln (3^{3 x + 2})

$\ln (5^{x - 2}) = \ln (3^{3 x + 2})$

Step 2

Expand

ln (5^{x - 2})

$\ln (5^{x - 2})$ by moving

x - 2

$x - 2$ outside the logarithm.

(x - 2) ln (5) = ln (3^{3 x + 2})

$(x - 2) \ln (5) = \ln (3^{3 x + 2})$

Step 3

Expand

ln (3^{3 x + 2})

$\ln (3^{3 x + 2})$ by moving

3 x + 2

$3 x + 2$ outside the logarithm.

(x - 2) ln (5) = (3 x + 2) ln (3)

$(x - 2) \ln (5) = (3 x + 2) \ln (3)$

Step 4

Solve the equation for

x

$x$ .

Tap for more steps...

Step 4.1

Simplify

(x - 2) ln (5)

$(x - 2) \ln (5)$ .

Tap for more steps...

Step 4.1.1

Rewrite.

0 + 0 + (x - 2) ln (5) = (3 x + 2) ln (3)

$0 + 0 + (x - 2) \ln (5) = (3 x + 2) \ln (3)$

Step 4.1.2

Simplify by adding zeros.

(x - 2) ln (5) = (3 x + 2) ln (3)

$(x - 2) \ln (5) = (3 x + 2) \ln (3)$

Step 4.1.3

Apply the distributive property.

x ln (5) - 2 ln (5) = (3 x + 2) ln (3)

$x \ln (5) - 2 \ln (5) = (3 x + 2) \ln (3)$

x ln (5) - 2 ln (5) = (3 x + 2) ln (3)

$x \ln (5) - 2 \ln (5) = (3 x + 2) \ln (3)$

Step 4.2

Apply the distributive property.

x ln (5) - 2 ln (5) = 3 x ln (3) + 2 ln (3)

$x \ln (5) - 2 \ln (5) = 3 x \ln (3) + 2 \ln (3)$

Step 4.3

Subtract

3 x ln (3)

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

x ln (5) - 2 ln (5) - 3 x ln (3) = 2 ln (3)

$x \ln (5) - 2 \ln (5) - 3 x \ln (3) = 2 \ln (3)$

Step 4.4

Add

2 ln (5)

$2 \ln (5)$ to both sides of the equation.

x ln (5) - 3 x ln (3) = 2 ln (3) + 2 ln (5)

$x \ln (5) - 3 x \ln (3) = 2 \ln (3) + 2 \ln (5)$

Step 4.5

Factor

x

$x$ out of

x ln (5) - 3 x ln (3)

$x \ln (5) - 3 x \ln (3)$ .

Tap for more steps...

Step 4.5.1

Factor

x

$x$ out of

x ln (5)

$x \ln (5)$ .

x (ln (5)) - 3 x ln (3) = 2 ln (3) + 2 ln (5)

$x (\ln (5)) - 3 x \ln (3) = 2 \ln (3) + 2 \ln (5)$

Step 4.5.2

Factor

x

$x$ out of

- 3 x ln (3)

$- 3 x \ln (3)$ .

x (ln (5)) + x (- 3 ln (3)) = 2 ln (3) + 2 ln (5)

$x (\ln (5)) + x (- 3 \ln (3)) = 2 \ln (3) + 2 \ln (5)$

Step 4.5.3

Factor

x

$x$ out of

x (ln (5)) + x (- 3 ln (3))

$x (\ln (5)) + x (- 3 \ln (3))$ .

x (ln (5) - 3 ln (3)) = 2 ln (3) + 2 ln (5)

$x (\ln (5) - 3 \ln (3)) = 2 \ln (3) + 2 \ln (5)$

x (ln (5) - 3 ln (3)) = 2 ln (3) + 2 ln (5)

$x (\ln (5) - 3 \ln (3)) = 2 \ln (3) + 2 \ln (5)$

Step 4.6

Divide each term in

x (ln (5) - 3 ln (3)) = 2 ln (3) + 2 ln (5)

$x (\ln (5) - 3 \ln (3)) = 2 \ln (3) + 2 \ln (5)$ by

ln (5) - 3 ln (3)

$\ln (5) - 3 \ln (3)$ and simplify.

Tap for more steps...

Step 4.6.1

Divide each term in

x (ln (5) - 3 ln (3)) = 2 ln (3) + 2 ln (5)

$x (\ln (5) - 3 \ln (3)) = 2 \ln (3) + 2 \ln (5)$ by

ln (5) - 3 ln (3)

$\ln (5) - 3 \ln (3)$ .

\frac{x (ln (5) - 3 ln (3))}{ln (5) - 3 ln (3)} = \frac{2 ln (3)}{ln (5) - 3 ln (3)} + \frac{2 ln (5)}{ln (5) - 3 ln (3)}

$\frac{x (\ln (5) - 3 \ln (3))}{\ln (5) - 3 \ln (3)} = \frac{2 \ln (3)}{\ln (5) - 3 \ln (3)} + \frac{2 \ln (5)}{\ln (5) - 3 \ln (3)}$

Step 4.6.2

Simplify the left side.

Tap for more steps...

Step 4.6.2.1

Cancel the common factor of

ln (5) - 3 ln (3)

$\ln (5) - 3 \ln (3)$ .

Tap for more steps...

Step 4.6.2.1.1

Cancel the common factor.

\frac{x (ln (5) - 3 ln (3))}{ln (5) - 3 ln (3)} = \frac{2 ln (3)}{ln (5) - 3 ln (3)} + \frac{2 ln (5)}{ln (5) - 3 ln (3)}

$\frac{x (\ln (5) - 3 \ln (3))}{\ln (5) - 3 \ln (3)} = \frac{2 \ln (3)}{\ln (5) - 3 \ln (3)} + \frac{2 \ln (5)}{\ln (5) - 3 \ln (3)}$

Step 4.6.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{2 ln (3)}{ln (5) - 3 ln (3)} + \frac{2 ln (5)}{ln (5) - 3 ln (3)}

$x = \frac{2 \ln (3)}{\ln (5) - 3 \ln (3)} + \frac{2 \ln (5)}{\ln (5) - 3 \ln (3)}$

x = \frac{2 ln (3)}{ln (5) - 3 ln (3)} + \frac{2 ln (5)}{ln (5) - 3 ln (3)}

$x = \frac{2 \ln (3)}{\ln (5) - 3 \ln (3)} + \frac{2 \ln (5)}{\ln (5) - 3 \ln (3)}$

x = \frac{2 ln (3)}{ln (5) - 3 ln (3)} + \frac{2 ln (5)}{ln (5) - 3 ln (3)}

$x = \frac{2 \ln (3)}{\ln (5) - 3 \ln (3)} + \frac{2 \ln (5)}{\ln (5) - 3 \ln (3)}$

Step 4.6.3

Simplify the right side.

Tap for more steps...

Step 4.6.3.1

Combine the numerators over the common denominator.

x = \frac{2 ln (3) + 2 ln (5)}{ln (5) - 3 ln (3)}

$x = \frac{2 \ln (3) + 2 \ln (5)}{\ln (5) - 3 \ln (3)}$

Step 4.6.3.2

Factor

2

$2$ out of

2 ln (3) + 2 ln (5)

$2 \ln (3) + 2 \ln (5)$ .

Tap for more steps...

Step 4.6.3.2.1

Factor

2

$2$ out of

2 ln (3)

$2 \ln (3)$ .

x = \frac{2 ln (3) + 2 ln (5)}{ln (5) - 3 ln (3)}

$x = \frac{2 \ln (3) + 2 \ln (5)}{\ln (5) - 3 \ln (3)}$

Step 4.6.3.2.2

Factor

2

$2$ out of

2 ln (5)

$2 \ln (5)$ .

x = \frac{2 ln (3) + 2 ln (5)}{ln (5) - 3 ln (3)}

$x = \frac{2 \ln (3) + 2 \ln (5)}{\ln (5) - 3 \ln (3)}$

Step 4.6.3.2.3

Factor

2

$2$ out of

2 ln (3) + 2 ln (5)

$2 \ln (3) + 2 \ln (5)$ .

x = \frac{2 (ln (3) + ln (5))}{ln (5) - 3 ln (3)}

$x = \frac{2 (\ln (3) + \ln (5))}{\ln (5) - 3 \ln (3)}$

x = \frac{2 (ln (3) + ln (5))}{ln (5) - 3 ln (3)}

$x = \frac{2 (\ln (3) + \ln (5))}{\ln (5) - 3 \ln (3)}$

x = \frac{2 (ln (3) + ln (5))}{ln (5) - 3 ln (3)}

$x = \frac{2 (\ln (3) + \ln (5))}{\ln (5) - 3 \ln (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 (\ln (3) + \ln (5))}{\ln (5) - 3 \ln (3)}$

Decimal Form:

$x = - 3.21163648 \dots$