Solve for d 5^-3*2^-3=10^d

$5^{- 3} \cdot 2^{- 3} = 10^{d}$

Step 1

Rewrite the equation as $10^{d} = 5^{- 3} \cdot 2^{- 3}$ .

$10^{d} = 5^{- 3} \cdot 2^{- 3}$

Step 2

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

$\log (10^{d}) = \log (5^{- 3} \cdot 2^{- 3})$

Step 3

Expand the left side.

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

$d \log (10) = \log (5^{- 3} \cdot 2^{- 3})$

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

$d \cdot 1 = \log (5^{- 3} \cdot 2^{- 3})$

Multiply $d$ by $1$ .

$d = \log (5^{- 3} \cdot 2^{- 3})$

Step 4

Simplify.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$d = \log (\frac{1}{5^{3}} \cdot 2^{- 3})$

Raise $5$ to the power of $3$ .

$d = \log (\frac{1}{125} \cdot 2^{- 3})$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$d = \log (\frac{1}{125} \cdot \frac{1}{2^{3}})$

Combine.

$d = \log (\frac{1 \cdot 1}{125 \cdot 2^{3}})$

Multiply $1$ by $1$ .

$d = \log (\frac{1}{125 \cdot 2^{3}})$

Raise $2$ to the power of $3$ .

$d = \log (\frac{1}{125 \cdot 8})$

Multiply $125$ by $8$ .

$d = \log (\frac{1}{1000})$

Logarithm base $10$ of $\frac{1}{1000}$ is $- 3$ .

$d = - 3$