Precalculus Examples

$15^{- 7 y} = 2$

Step 1

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

$\ln (15^{- 7 y}) = \ln (2)$

Step 2

Expand $\ln (15^{- 7 y})$ by moving $- 7 y$ outside the logarithm.

$- 7 y \ln (15) = \ln (2)$

Step 3

Divide each term in $- 7 y \ln (15) = \ln (2)$ by $- 7 \ln (15)$ and simplify.

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Step 3.1

Divide each term in $- 7 y \ln (15) = \ln (2)$ by $- 7 \ln (15)$ .

$\frac{- 7 y \ln (15)}{- 7 \ln (15)} = \frac{\ln (2)}{- 7 \ln (15)}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 7$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 7 y \ln (15)}{- 7 \ln (15)} = \frac{\ln (2)}{- 7 \ln (15)}$

Step 3.2.1.2

Rewrite the expression.

$\frac{y \ln (15)}{\ln (15)} = \frac{\ln (2)}{- 7 \ln (15)}$

Step 3.2.2

Cancel the common factor of $\ln (15)$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{y \ln (15)}{\ln (15)} = \frac{\ln (2)}{- 7 \ln (15)}$

Step 3.2.2.2

Divide $y$ by $1$ .

$y = \frac{\ln (2)}{- 7 \ln (15)}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$y = - \frac{\ln (2)}{7 \ln (15)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{\ln (2)}{7 \ln (15)}$

Decimal Form:

$y = - 0.03656543 \dots$