Evaluate: (log_{10}((3.5 * 10^3)-(1.7 * 10^3)))/(30-40)

Expression: $\frac{ \log_{ 10 }({ \left( 3.5 \times {10}^{3} \right)-\left( 1.7 \times {10}^{3} \right) }) }{ 30-40 }$

Remove unnecessary parentheses

$\frac{ \log_{ 10 }({ 3.5 \times {10}^{3}-\left( 1.7 \times {10}^{3} \right) }) }{ 30-40 }$

Remove unnecessary parentheses

$\frac{ \log_{ 10 }({ 3.5 \times {10}^{3}-1.7 \times {10}^{3} }) }{ 30-40 }$

Calculate the difference

$\frac{ \log_{ 10 }({ 3.5 \times {10}^{3}-1.7 \times {10}^{3} }) }{ -10 }$

Collect like terms

$\frac{ \log_{ 10 }({ 1.8 \times {10}^{3} }) }{ -10 }$

Use $\log_{ a }({ x \times y })=\log_{ a }({ x })+\log_{ a }({ y })$ to expand the expression

$\frac{ \log_{ 10 }({ 1.8 })+\log_{ 10 }({ {10}^{3} }) }{ -10 }$

Convert the decimal into a fraction

$\frac{ \log_{ 10 }({ \frac{ 18 }{ 10 } })+\log_{ 10 }({ {10}^{3} }) }{ -10 }$

Use $\log_{ a }({ {a}^{x} })=x$ to simplify the expression

$\frac{ \log_{ 10 }({ \frac{ 18 }{ 10 } })+3 }{ -10 }$

Use $\log_{ a }({ \frac{ x }{ y } })=\log_{ a }({ x })-\log_{ a }({ y })$ to expand the expression

$\frac{ \log_{ 10 }({ 18 })-\log_{ 10 }({ 10 })+3 }{ -10 }$

A logarithm with the same base and argument equals $1$

$\frac{ \log_{ 10 }({ 18 })-1+3 }{ -10 }$

Calculate the sum

$\frac{ \log_{ 10 }({ 18 })+2 }{ -10 }$

Use $\frac{ -a }{ b }=\frac{ a }{ -b }=-\frac{ a }{ b }$ to rewrite the fraction

\begin{align*}&-\frac{ \log_{ 10 }({ 18 })+2 }{ 10 } \\&\approx-0.325527\end{align*}

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