Trigonometry Examples

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

Step 1

Take the log of both sides of the equation.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x - 1) \ln (5)$ .

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

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

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

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

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

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

Step 4.3

Subtract $2 x \ln (3)$ from both sides of the equation.

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

Step 4.4

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

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

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

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Step 4.6.2.1.1

Cancel the common factor.

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

Step 4.6.2.1.2

Divide $x$ by $1$ .

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

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Combine the numerators over the common denominator.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 4.60719911 \dots$