Precalculus Examples

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

Step 1

Take the log of both sides of the equation.

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

Step 2

Expand $\ln (5^{3 x + 1})$ by moving $3 x + 1$ outside the logarithm.

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

Step 3

Expand $\ln (3^{x - 5})$ by moving $x - 5$ outside the logarithm.

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

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(3 x + 1) \ln (5)$ .

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

Multiply $\ln (5)$ by $1$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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

Step 4.6

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

Simplify the left side.

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Step 4.7.2.1

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

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Step 4.7.2.1.1

Cancel the common factor.

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

Step 4.7.2.1.2

Divide $x$ by $1$ .

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

Step 4.7.3

Simplify the right side.

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Step 4.7.3.1

Combine the numerators over the common denominator.

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

Step 4.7.3.2

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

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

Step 4.7.3.3

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

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

Step 4.7.3.4

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

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

Step 4.7.3.5

Simplify the expression.

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Step 4.7.3.5.1

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

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

Step 4.7.3.5.2

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.90430774 \dots$