Calculate: (log_{10}((18 * 10^4)-(3.2 * 10^4)))/(30-40)

Expression: $\frac{ \log_{ 10 }({ \left( 18 \times {10}^{4} \right)-\left( 3.2 \times {10}^{4} \right) }) }{ 30-40 }$

Remove unnecessary parentheses

$\frac{ \log_{ 10 }({ 18 \times {10}^{4}-\left( 3.2 \times {10}^{4} \right) }) }{ 30-40 }$

Remove unnecessary parentheses

$\frac{ \log_{ 10 }({ 18 \times {10}^{4}-3.2 \times {10}^{4} }) }{ 30-40 }$

Calculate the difference

$\frac{ \log_{ 10 }({ 18 \times {10}^{4}-3.2 \times {10}^{4} }) }{ -10 }$

Collect like terms

$\frac{ \log_{ 10 }({ 14.8 \times {10}^{4} }) }{ -10 }$

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

$\frac{ \log_{ 10 }({ 14.8 })+\log_{ 10 }({ {10}^{4} }) }{ -10 }$

Convert the decimal into a fraction

$\frac{ \log_{ 10 }({ \frac{ 148 }{ 10 } })+\log_{ 10 }({ {10}^{4} }) }{ -10 }$

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

$\frac{ \log_{ 10 }({ \frac{ 148 }{ 10 } })+4 }{ -10 }$

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

$\frac{ \log_{ 10 }({ 148 })-\log_{ 10 }({ 10 })+4 }{ -10 }$

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

$\frac{ \log_{ 10 }({ 148 })-1+4 }{ -10 }$

Calculate the sum

$\frac{ \log_{ 10 }({ 148 })+3 }{ -10 }$

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

$\begin{align*}&-\frac{ \log_{ 10 }({ 148 })+3 }{ 10 } \\&\approx-0.517026\end{align*}$