Precalculus Examples

${1.2}^{x} = {0.5}^{- x}$

Step 1

Take the log of both sides of the equation.

$\ln ({1.2}^{x}) = \ln ({0.5}^{- x})$

Step 2

Expand $\ln ({1.2}^{x})$ by moving $x$ outside the logarithm.

$x \ln (1.2) = \ln ({0.5}^{- x})$

Step 3

Expand $\ln ({0.5}^{- x})$ by moving $- x$ outside the logarithm.

$x \ln (1.2) = - x \ln (0.5)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Add $x \ln (0.5)$ to both sides of the equation.

$x \ln (1.2) + x \ln (0.5) = 0$

Step 4.2

Factor $x$ out of $x \ln (1.2) + x \ln (0.5)$ .

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Step 4.2.1

Factor $x$ out of $x \ln (1.2)$ .

$x (\ln (1.2)) + x \ln (0.5) = 0$

Step 4.2.2

Factor $x$ out of $x \ln (0.5)$ .

$x (\ln (1.2)) + x (\ln (0.5)) = 0$

Step 4.2.3

Factor $x$ out of $x (\ln (1.2)) + x (\ln (0.5))$ .

$x (\ln (1.2) + \ln (0.5)) = 0$

Step 4.3

Divide each term in $x (\ln (1.2) + \ln (0.5)) = 0$ by $\ln (1.2) + \ln (0.5)$ and simplify.

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Step 4.3.1

Divide each term in $x (\ln (1.2) + \ln (0.5)) = 0$ by $\ln (1.2) + \ln (0.5)$ .

$\frac{x (\ln (1.2) + \ln (0.5))}{\ln (1.2) + \ln (0.5)} = \frac{0}{\ln (1.2) + \ln (0.5)}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $\ln (1.2) + \ln (0.5)$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{x (\ln (1.2) + \ln (0.5))}{\ln (1.2) + \ln (0.5)} = \frac{0}{\ln (1.2) + \ln (0.5)}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{\ln (1.2) + \ln (0.5)}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $0$ by $\ln (1.2) + \ln (0.5)$ .

$x = 0$