Precalculus Examples

$5^{3 x - 1} = 4^{2 - x}$

Step 1

Take the log of both sides of the equation.

$\ln (5^{3 x - 1}) = \ln (4^{2 - x})$

Step 2

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

$(3 x - 1) \ln (5) = \ln (4^{2 - x})$

Step 3

Expand $\ln (4^{2 - x})$ by moving $2 - x$ outside the logarithm.

$(3 x - 1) \ln (5) = (2 - x) \ln (4)$

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.

$0 + 0 + (3 x - 1) \ln (5) = (2 - x) \ln (4)$

Step 4.1.2

Simplify by adding zeros.

$(3 x - 1) \ln (5) = (2 - x) \ln (4)$

Step 4.1.3

Apply the distributive property.

$3 x \ln (5) - 1 \ln (5) = (2 - x) \ln (4)$

Step 4.1.4

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

$3 x \ln (5) - \ln (5) = (2 - x) \ln (4)$

Step 4.2

Apply the distributive property.

$3 x \ln (5) - \ln (5) = 2 \ln (4) - x \ln (4)$

Step 4.3

Add $x \ln (4)$ to both sides of the equation.

$3 x \ln (5) - \ln (5) + x \ln (4) = 2 \ln (4)$

Step 4.4

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

$3 x \ln (5) + x \ln (4) = 2 \ln (4) + \ln (5)$

Step 4.5

Factor $x$ out of $3 x \ln (5) + x \ln (4)$ .

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Step 4.5.1

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

$x (3 \ln (5)) + x \ln (4) = 2 \ln (4) + \ln (5)$

Step 4.5.2

Factor $x$ out of $x \ln (4)$ .

$x (3 \ln (5)) + x (\ln (4)) = 2 \ln (4) + \ln (5)$

Step 4.5.3

Factor $x$ out of $x (3 \ln (5)) + x (\ln (4))$ .

$x (3 \ln (5) + \ln (4)) = 2 \ln (4) + \ln (5)$

Step 4.6

Divide each term in $x (3 \ln (5) + \ln (4)) = 2 \ln (4) + \ln (5)$ by $3 \ln (5) + \ln (4)$ and simplify.

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Step 4.6.1

Divide each term in $x (3 \ln (5) + \ln (4)) = 2 \ln (4) + \ln (5)$ by $3 \ln (5) + \ln (4)$ .

$\frac{x (3 \ln (5) + \ln (4))}{3 \ln (5) + \ln (4)} = \frac{2 \ln (4)}{3 \ln (5) + \ln (4)} + \frac{\ln (5)}{3 \ln (5) + \ln (4)}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $3 \ln (5) + \ln (4)$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{x (3 \ln (5) + \ln (4))}{3 \ln (5) + \ln (4)} = \frac{2 \ln (4)}{3 \ln (5) + \ln (4)} + \frac{\ln (5)}{3 \ln (5) + \ln (4)}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 \ln (4)}{3 \ln (5) + \ln (4)} + \frac{\ln (5)}{3 \ln (5) + \ln (4)}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Combine the numerators over the common denominator.

$x = \frac{2 \ln (4) + \ln (5)}{3 \ln (5) + \ln (4)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 \ln (4) + \ln (5)}{3 \ln (5) + \ln (4)}$

Decimal Form:

$x = 0.70511713 \dots$