# Calculate: 2log_{10}(x)-log_{10}(7x)-1=0

## Expression: $2\log_{ 10 }({ x })-\log_{ 10 }({ 7x })-1=0$

Determine the defined range

$\begin{array} { l }2\log_{ 10 }({ x })-\log_{ 10 }({ 7x })-1=0,& x \in \langle0, +\infty\rangle\end{array}$

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

$2\log_{ 10 }({ x })-\left( \log_{ 10 }({ 7 })+\log_{ 10 }({ x }) \right)-1=0$

When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses

$2\log_{ 10 }({ x })-\log_{ 10 }({ 7 })-\log_{ 10 }({ x })-1=0$

Collect like terms

$\log_{ 10 }({ x })-\log_{ 10 }({ 7 })-1=0$

Move the constants to the right-hand side and change their signs

$\log_{ 10 }({ x })=\log_{ 10 }({ 7 })+1$

$1$ can be expressed as a logarithm with the same base and argument

$\log_{ 10 }({ x })=\log_{ 10 }({ 7 })+\log_{ 10 }({ 10 })$

Calculate the sum of logarithms

$\log_{ 10 }({ x })=\log_{ 10 }({ 70 })$

Since the bases of the logarithms are the same, set the arguments equal

$\begin{array} { l }x=70,& x \in \langle0, +\infty\rangle\end{array}$

Check if the solution is in the defined range

$x=70$

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