Trigonometry Examples

$6^{7 x - 4} = 3^{2 x + 1}$

Step 1

Take the log of both sides of the equation.

$\ln (6^{7 x - 4}) = \ln (3^{2 x + 1})$

Step 2

Expand $\ln (6^{7 x - 4})$ by moving $7 x - 4$ outside the logarithm.

$(7 x - 4) \ln (6) = \ln (3^{2 x + 1})$

Step 3

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

$(7 x - 4) \ln (6) = (2 x + 1) \ln (3)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(7 x - 4) \ln (6)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (7 x - 4) \ln (6) = (2 x + 1) \ln (3)$

Step 4.1.2

Simplify by adding zeros.

$(7 x - 4) \ln (6) = (2 x + 1) \ln (3)$

Step 4.1.3

Apply the distributive property.

$7 x \ln (6) - 4 \ln (6) = (2 x + 1) \ln (3)$

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

$7 x \ln (6) - 4 \ln (6) = 2 x \ln (3) + 1 \ln (3)$

Step 4.2.2

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

$7 x \ln (6) - 4 \ln (6) = 2 x \ln (3) + \ln (3)$

Step 4.3

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

$7 x \ln (6) - 4 \ln (6) - 2 x \ln (3) = \ln (3)$

Step 4.4

Add $4 \ln (6)$ to both sides of the equation.

$7 x \ln (6) - 2 x \ln (3) = \ln (3) + 4 \ln (6)$

Step 4.5

Factor $x$ out of $7 x \ln (6) - 2 x \ln (3)$ .

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Step 4.5.1

Factor $x$ out of $7 x \ln (6)$ .

$x (7 \ln (6)) - 2 x \ln (3) = \ln (3) + 4 \ln (6)$

Step 4.5.2

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

$x (7 \ln (6)) + x (- 2 \ln (3)) = \ln (3) + 4 \ln (6)$

Step 4.5.3

Factor $x$ out of $x (7 \ln (6)) + x (- 2 \ln (3))$ .

$x (7 \ln (6) - 2 \ln (3)) = \ln (3) + 4 \ln (6)$

Step 4.6

Divide each term in $x (7 \ln (6) - 2 \ln (3)) = \ln (3) + 4 \ln (6)$ by $7 \ln (6) - 2 \ln (3)$ and simplify.

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Step 4.6.1

Divide each term in $x (7 \ln (6) - 2 \ln (3)) = \ln (3) + 4 \ln (6)$ by $7 \ln (6) - 2 \ln (3)$ .

$\frac{x (7 \ln (6) - 2 \ln (3))}{7 \ln (6) - 2 \ln (3)} = \frac{\ln (3)}{7 \ln (6) - 2 \ln (3)} + \frac{4 \ln (6)}{7 \ln (6) - 2 \ln (3)}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $7 \ln (6) - 2 \ln (3)$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{x (7 \ln (6) - 2 \ln (3))}{7 \ln (6) - 2 \ln (3)} = \frac{\ln (3)}{7 \ln (6) - 2 \ln (3)} + \frac{4 \ln (6)}{7 \ln (6) - 2 \ln (3)}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (3)}{7 \ln (6) - 2 \ln (3)} + \frac{4 \ln (6)}{7 \ln (6) - 2 \ln (3)}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Combine the numerators over the common denominator.

$x = \frac{\ln (3) + 4 \ln (6)}{7 \ln (6) - 2 \ln (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (3) + 4 \ln (6)}{7 \ln (6) - 2 \ln (3)}$

Decimal Form:

$x = 0.79899244 \dots$