Trigonometry Examples

$4^{4 x - 3} = 6^{- x - 1}$

Step 1

Take the log of both sides of the equation.

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

Step 2

Expand $\ln (4^{4 x - 3})$ by moving $4 x - 3$ outside the logarithm.

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

Step 3

Expand $\ln (6^{- x - 1})$ by moving $- x - 1$ outside the logarithm.

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

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(4 x - 3) \ln (4)$ .

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify $(- x - 1) \ln (6)$ .

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Rewrite $- 1 \ln (6)$ as $- \ln (6)$ .

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

Step 4.3

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

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

Step 4.4

Add $3 \ln (4)$ to both sides of the equation.

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

Step 4.5

Factor $x$ out of $4 x \ln (4) + x \ln (6)$ .

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

Factor $x$ out of $x (4 \ln (4)) + x (\ln (6))$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $4 \ln (4) + \ln (6)$ .

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Step 4.6.2.1.1

Cancel the common factor.

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

Step 4.6.2.1.2

Divide $x$ by $1$ .

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

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Combine the numerators over the common denominator.

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

Step 4.6.3.2

Factor $- 1$ out of $- \ln (6)$ .

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

Step 4.6.3.3

Factor $- 1$ out of $3 \ln (4)$ .

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

Step 4.6.3.4

Factor $- 1$ out of $- \ln (6) - (- 3 \ln (4))$ .

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

Step 4.6.3.5

Simplify the expression.

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Step 4.6.3.5.1

Rewrite $- (\ln (6) - 3 \ln (4))$ as $- 1 (\ln (6) - 3 \ln (4))$ .

$x = \frac{- 1 (\ln (6) - 3 \ln (4))}{4 \ln (4) + \ln (6)}$

Step 4.6.3.5.2

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.32263104 \dots$