Trigonometry Examples

${41.2}^{x} = {12.6}^{x - 1}$

Step 1

Take the log of both sides of the equation.

$\ln ({41.2}^{x}) = \ln ({12.6}^{x - 1})$

Step 2

Expand $\ln ({41.2}^{x})$ by moving $x$ outside the logarithm.

$x \ln (41.2) = \ln ({12.6}^{x - 1})$

Step 3

Expand $\ln ({12.6}^{x - 1})$ by moving $x - 1$ outside the logarithm.

$x \ln (41.2) = (x - 1) \ln (12.6)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x - 1) \ln (12.6)$ .

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Step 4.1.1

Apply the distributive property.

$x \ln (41.2) = x \ln (12.6) - 1 \ln (12.6)$

Step 4.1.2

Rewrite $- 1 \ln (12.6)$ as $- \ln (12.6)$ .

$x \ln (41.2) = x \ln (12.6) - \ln (12.6)$

Step 4.2

Subtract $x \ln (12.6)$ from both sides of the equation.

$x \ln (41.2) - x \ln (12.6) = - \ln (12.6)$

Step 4.3

Factor $x$ out of $x \ln (41.2) - x \ln (12.6)$ .

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Step 4.3.1

Factor $x$ out of $x \ln (41.2)$ .

$x (\ln (41.2)) - x \ln (12.6) = - \ln (12.6)$

Step 4.3.2

Factor $x$ out of $- x \ln (12.6)$ .

$x (\ln (41.2)) + x (- 1 \ln (12.6)) = - \ln (12.6)$

Step 4.3.3

Factor $x$ out of $x (\ln (41.2)) + x (- 1 \ln (12.6))$ .

$x (\ln (41.2) - 1 \ln (12.6)) = - \ln (12.6)$

Step 4.4

Rewrite $- 1 \ln (12.6)$ as $- \ln (12.6)$ .

$x (\ln (41.2) - \ln (12.6)) = - \ln (12.6)$

Step 4.5

Divide each term in $x (\ln (41.2) - \ln (12.6)) = - \ln (12.6)$ by $\ln (41.2) - \ln (12.6)$ and simplify.

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Step 4.5.1

Divide each term in $x (\ln (41.2) - \ln (12.6)) = - \ln (12.6)$ by $\ln (41.2) - \ln (12.6)$ .

$\frac{x (\ln (41.2) - \ln (12.6))}{\ln (41.2) - \ln (12.6)} = \frac{- \ln (12.6)}{\ln (41.2) - \ln (12.6)}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $\ln (41.2) - \ln (12.6)$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{x (\ln (41.2) - \ln (12.6))}{\ln (41.2) - \ln (12.6)} = \frac{- \ln (12.6)}{\ln (41.2) - \ln (12.6)}$

Step 4.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (12.6)}{\ln (41.2) - \ln (12.6)}$

Step 4.5.3

Simplify the right side.

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Step 4.5.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (12.6)}{\ln (41.2) - \ln (12.6)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (12.6)}{\ln (41.2) - \ln (12.6)}$

Decimal Form:

$x = - 2.13860739 \dots$