Solve for y 2*10^(5-y)+9=87

$2 \cdot 10^{5 - y} + 9 = 87$

Step 1

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

Subtract $9$ from both sides of the equation.

$2 \cdot 10^{5 - y} = 87 - 9$

Subtract $9$ from $87$ .

$2 \cdot 10^{5 - y} = 78$

Step 2

Divide each term in $(2 \cdot 10^{5 - y}) = 78$ by $2$ and simplify.

Divide each term in $(2 \cdot 10^{5 - y}) = 78$ by $2$ .

$\frac{(2 \cdot 10^{5 - y})}{2} = \frac{78}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \cdot 10^{5 - y}}{2} = \frac{78}{2}$

Divide $10^{5 - y}$ by $1$ .

$10^{5 - y} = \frac{78}{2}$

Simplify the right side.

Divide $78$ by $2$ .

$10^{5 - y} = 39$

Step 3

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{5 - y}) = \log (39)$

Step 4

Expand the left side.

Expand $\log (10^{5 - y})$ by moving $5 - y$ outside the logarithm.

$(5 - y) \log (10) = \log (39)$

Logarithm base $10$ of $10$ is $1$ .

$(5 - y) \cdot 1 = \log (39)$

Multiply $5 - y$ by $1$ .

$5 - y = \log (39)$

Step 5

Subtract $5$ from both sides of the equation.

$- y = \log (39) - 5$

Step 6

Divide each term in $- y = \log (39) - 5$ by $- 1$ and simplify.

Divide each term in $- y = \log (39) - 5$ by $- 1$ .

$\frac{- y}{- 1} = \frac{\log (39)}{- 1} + \frac{- 5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{\log (39)}{- 1} + \frac{- 5}{- 1}$

Divide $y$ by $1$ .

$y = \frac{\log (39)}{- 1} + \frac{- 5}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{\log (39)}{- 1}$ .

$y = - 1 \cdot \log (39) + \frac{- 5}{- 1}$

Rewrite $- 1 \cdot \log (39)$ as $- \log (39)$ .

$y = - \log (39) + \frac{- 5}{- 1}$

Divide $- 5$ by $- 1$ .

$y = - \log (39) + 5$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = - \log (39) + 5$

Decimal Form:

$y = 3.40893539 \dots$

Solve for y 2*10^(5-y)+9=87

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