Precalculus Examples

$8^{y - 3} = 11$

Step 1

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

$\ln (8^{y - 3}) = \ln (11)$

Step 2

Expand $\ln (8^{y - 3})$ by moving $y - 3$ outside the logarithm.

$(y - 3) \ln (8) = \ln (11)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$y \ln (8) - 3 \ln (8) = \ln (11)$

Step 4

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

$y \ln (8) - 3 \ln (8) - \ln (11) = 0$

Step 5

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

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Step 5.1

Add $3 \ln (8)$ to both sides of the equation.

$y \ln (8) - \ln (11) = 3 \ln (8)$

Step 5.2

Add $\ln (11)$ to both sides of the equation.

$y \ln (8) = 3 \ln (8) + \ln (11)$

Step 6

Divide each term in $y \ln (8) = 3 \ln (8) + \ln (11)$ by $\ln (8)$ and simplify.

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Step 6.1

Divide each term in $y \ln (8) = 3 \ln (8) + \ln (11)$ by $\ln (8)$ .

$\frac{y \ln (8)}{\ln (8)} = \frac{3 \ln (8)}{\ln (8)} + \frac{\ln (11)}{\ln (8)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\ln (8)$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{y \ln (8)}{\ln (8)} = \frac{3 \ln (8)}{\ln (8)} + \frac{\ln (11)}{\ln (8)}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{3 \ln (8)}{\ln (8)} + \frac{\ln (11)}{\ln (8)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $\ln (8)$ .

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Step 6.3.1.1

Cancel the common factor.

$y = \frac{3 \ln (8)}{\ln (8)} + \frac{\ln (11)}{\ln (8)}$

Step 6.3.1.2

Divide $3$ by $1$ .

$y = 3 + \frac{\ln (11)}{\ln (8)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = 3 + \frac{\ln (11)}{\ln (8)}$

Decimal Form:

$y = 4.15314387 \dots$