$\log_{ 10 }({ \frac{ 75 }{ 10 } })+\log_{ 10 }({ {\left( \frac{ 5 }{ 9 } \right)}^{-2} })+\log_{ 10 }({ \frac{ 100 }{ 243 } })$
Use the logarithmic product and quotient rules to simplify the expression$\log_{ 10 }({ \frac{ 75 }{ 10 } \times {\left( \frac{ 5 }{ 9 } \right)}^{-2} \times \frac{ 100 }{ 243 } })$
Cancel out the greatest common factor $10$$\log_{ 10 }({ 75 \times {\left( \frac{ 5 }{ 9 } \right)}^{-2} \times \frac{ 10 }{ 243 } })$
Cancel out the greatest common factor $3$$\log_{ 10 }({ 25 \times {\left( \frac{ 5 }{ 9 } \right)}^{-2} \times \frac{ 10 }{ 81 } })$
To raise a fraction to a power, raise the numerator and denominator to that power$\log_{ 10 }({ 25 \times \frac{ {5}^{-2} }{ {9}^{-2} } \times \frac{ 10 }{ 81 } })$
Calculate the product$\log_{ 10 }({ \frac{ 250 \times {5}^{-2} }{ 81 \times {9}^{-2} } })$
Express with a positive exponent using ${a}^{-n}=\frac{ 1 }{ {a}^{n} }$$\log_{ 10 }({ \frac{ 250 \times \frac{ 1 }{ {5}^{2} } }{ 81 \times {9}^{-2} } })$
Write the number in exponential form with the base of $9$$\log_{ 10 }({ \frac{ 250 \times \frac{ 1 }{ {5}^{2} } }{ {9}^{2} \times {9}^{-2} } })$
Evaluate the power$\log_{ 10 }({ \frac{ 250 \times \frac{ 1 }{ 25 } }{ {9}^{2} \times {9}^{-2} } })$
Calculate the product$\log_{ 10 }({ \frac{ 250 \times \frac{ 1 }{ 25 } }{ 1 } })$
Any expression divided by $1$ remains the same$\log_{ 10 }({ 250 \times \frac{ 1 }{ 25 } })$
Cancel out the greatest common factor $25$$\log_{ 10 }({ 10 })$
A logarithm with the same base and argument equals $1$$1$