Trigonometry Examples

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

Step 1

Take the log of both sides of the equation.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve the equation for $x$ .

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Step 4.1

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

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

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

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

Step 4.2

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

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

Step 4.3

Add $\ln (3)$ to both sides of the equation.

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.5

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

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

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

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Step 4.6.2.1.1

Cancel the common factor.

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

Step 4.6.2.1.2

Divide $x$ by $1$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.86906637 \dots$