Precalculus Examples

$3^{1 - 2 x} = 5^{x + 5}$

Step 1

Take the log of both sides of the equation.

$\ln (3^{1 - 2 x}) = \ln (5^{x + 5})$

Step 2

Expand $\ln (3^{1 - 2 x})$ by moving $1 - 2 x$ outside the logarithm.

$(1 - 2 x) \ln (3) = \ln (5^{x + 5})$

Step 3

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

$(1 - 2 x) \ln (3) = (x + 5) \ln (5)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(1 - 2 x) \ln (3)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (1 - 2 x) \ln (3) = (x + 5) \ln (5)$

Step 4.1.2

Simplify by adding zeros.

$(1 - 2 x) \ln (3) = (x + 5) \ln (5)$

Step 4.1.3

Apply the distributive property.

$1 \ln (3) - 2 x \ln (3) = (x + 5) \ln (5)$

Step 4.1.4

Multiply $\ln (3)$ by $1$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Factor $x$ out of $- 2 x \ln (3) - x \ln (5)$ .

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Step 4.5.1

Factor $x$ out of $- 2 x \ln (3)$ .

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

Step 4.5.2

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

$x (- 2 \ln (3)) + x (- 1 \ln (5)) = 5 \ln (5) - \ln (3)$

Step 4.5.3

Factor $x$ out of $x (- 2 \ln (3)) + x (- 1 \ln (5))$ .

$x (- 2 \ln (3) - 1 \ln (5)) = 5 \ln (5) - \ln (3)$

Step 4.6

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

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

Step 4.7

Divide each term in $x (- 2 \ln (3) - \ln (5)) = 5 \ln (5) - \ln (3)$ by $- 2 \ln (3) - \ln (5)$ and simplify.

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Step 4.7.1

Divide each term in $x (- 2 \ln (3) - \ln (5)) = 5 \ln (5) - \ln (3)$ by $- 2 \ln (3) - \ln (5)$ .

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

Step 4.7.2

Simplify the left side.

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Step 4.7.2.1

Cancel the common factor of $- 2 \ln (3) - \ln (5)$ .

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Step 4.7.2.1.1

Cancel the common factor.

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

Step 4.7.2.1.2

Divide $x$ by $1$ .

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

Step 4.7.3

Simplify the right side.

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Step 4.7.3.1

Combine the numerators over the common denominator.

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

Step 4.7.3.2

Factor $- 1$ out of $- 2 \ln (3)$ .

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

Step 4.7.3.3

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

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

Step 4.7.3.4

Factor $- 1$ out of $- (2 \ln (3)) - \ln (5)$ .

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

Step 4.7.3.5

Rewrite negatives.

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Step 4.7.3.5.1

Rewrite $- (2 \ln (3) + \ln (5))$ as $- 1 (2 \ln (3) + \ln (5))$ .

$x = \frac{5 \ln (5) - \ln (3)}{- 1 (2 \ln (3) + \ln (5))}$

Step 4.7.3.5.2

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.82537256 \dots$