Solve for: log_{10}((75)/(10))-2log_{10}((5)/(9))+log_{10}((100)/(243))

Expression: $\log_{ 10 }({ \frac{ 75 }{ 10 } })-2\log_{ 10 }({ \frac{ 5 }{ 9 } })+\log_{ 10 }({ \frac{ 100 }{ 243 } })$

Use $x \times \log_{ a }({ b })=\log_{ a }({ {b}^{x} })$ to transform the expression

$\log_{ 10 }({ \frac{ 75 }{ 10 } })+\log_{ 10 }({ {\left( \frac{ 5 }{ 9 } \right)}^{-2} })+\log_{ 10 }({ \frac{ 100 }{ 243 } })$

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 $1$

$1$