Pre-Algebra Examples

$6^{x - 4} = y^{3 + 66}$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (6^{x - 4}) = \ln (y^{3 + 66})$

Step 2

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

$(x - 4) \ln (6) = \ln (y^{3 + 66})$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$x \ln (6) - 4 \ln (6) = \ln (y^{3 + 66})$

Step 4

Simplify the right side.

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Step 4.1

Add $3$ and $66$ .

$x \ln (6) - 4 \ln (6) = \ln (y^{69})$

Step 5

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (6) - 4 \ln (6) - \ln (y^{69}) = 0$

Step 6

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1

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

$x \ln (6) - \ln (y^{69}) = 4 \ln (6)$

Step 6.2

Add $\ln (y^{69})$ to both sides of the equation.

$x \ln (6) = 4 \ln (6) + \ln (y^{69})$

Step 7

Divide each term in $x \ln (6) = 4 \ln (6) + \ln (y^{69})$ by $\ln (6)$ and simplify.

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Step 7.1

Divide each term in $x \ln (6) = 4 \ln (6) + \ln (y^{69})$ by $\ln (6)$ .

$\frac{x \ln (6)}{\ln (6)} = \frac{4 \ln (6)}{\ln (6)} + \frac{\ln (y^{69})}{\ln (6)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{x \ln (6)}{\ln (6)} = \frac{4 \ln (6)}{\ln (6)} + \frac{\ln (y^{69})}{\ln (6)}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{4 \ln (6)}{\ln (6)} + \frac{\ln (y^{69})}{\ln (6)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

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

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Step 7.3.1.1

Cancel the common factor.

$x = \frac{4 \ln (6)}{\ln (6)} + \frac{\ln (y^{69})}{\ln (6)}$

Step 7.3.1.2

Divide $4$ by $1$ .

$x = 4 + \frac{\ln (y^{69})}{\ln (6)}$