Solve the equation
[math]9 \log_{ x }{ 5 } = \log_{ 5 }{ x }[/math]
we have
[math]\log_{ x }{ 5 ^ { 9 }} = \frac{ \log_{ 5 }{ 5 ^ { 9 }} }{ \log_{ 5 }{ x } }[/math]Equating it to the right, we have
[math]\frac{ \log_{ 5 }{ 5 ^ { 9 }} }{ \log_{ 5 }{ x } }= \log_{ 5 }{ x }[/math] [math]\frac{ \log_{ 5 }{ 5 ^ { 9 }} }{ \log_{ 5 }{ x } }= \frac{ \log_{ 5 }{ x } }{ 1 }[/math]
Cross multiplying, we have
[math]\log_{ 5 }{ 5 ^ { 9 }} = \log_{ 5 }{ x } \times \log_{ 5 }{ x }[/math]I am stucked here. What am I doing right?
If I choose to go the other way, that is changing the base of the right term,
[math]9 \log_{ x }{ 5 } = \log_{ 5 }{ x }[/math][math]\log_{ 5 }{ x } = \frac{ \log_{ x }{ x } }{ \log_{ x }{ 5 } }[/math][math]\frac{ 9 \log_{ x }{ 5 } }{ 1 } = \frac{ \log_{ x }{ x } }{ \log_{ x }{ 5 } }[/math][math]\frac{ 9 \log_{ x }{ 5 } }{ 1 } = \frac{ 1 }{ \log_{ x }{ 5 } }[/math][math]1 \times 1 = \log_{ x }{ 5 ^ { 9 }} \times \log_{ x }{ 5 }[/math] [math]\log_{ x }{ 5 ^ { 9 }} \left( \log_{ x }{ 5 } \right) = 1[/math]I am still stucked. I think need help.
[math]9 \log_{ x }{ 5 } = \log_{ 5 }{ x }[/math]
Solution
[math]\log_{ x } { 5 }^{9} = \log_{ 5 }{ x }[/math]Changing the base of the left term,we have
[math]\log_{ x }{ 5 ^ { 9 }} = \frac{ \log_{ 5 }{ 5 ^ { 9 }} }{ \log_{ 5 }{ x } }[/math]Equating it to the right, we have
[math]\frac{ \log_{ 5 }{ 5 ^ { 9 }} }{ \log_{ 5 }{ x } }= \log_{ 5 }{ x }[/math] [math]\frac{ \log_{ 5 }{ 5 ^ { 9 }} }{ \log_{ 5 }{ x } }= \frac{ \log_{ 5 }{ x } }{ 1 }[/math]
Cross multiplying, we have
[math]\log_{ 5 }{ 5 ^ { 9 }} = \log_{ 5 }{ x } \times \log_{ 5 }{ x }[/math]I am stucked here. What am I doing right?
If I choose to go the other way, that is changing the base of the right term,
[math]9 \log_{ x }{ 5 } = \log_{ 5 }{ x }[/math][math]\log_{ 5 }{ x } = \frac{ \log_{ x }{ x } }{ \log_{ x }{ 5 } }[/math][math]\frac{ 9 \log_{ x }{ 5 } }{ 1 } = \frac{ \log_{ x }{ x } }{ \log_{ x }{ 5 } }[/math][math]\frac{ 9 \log_{ x }{ 5 } }{ 1 } = \frac{ 1 }{ \log_{ x }{ 5 } }[/math][math]1 \times 1 = \log_{ x }{ 5 ^ { 9 }} \times \log_{ x }{ 5 }[/math] [math]\log_{ x }{ 5 ^ { 9 }} \left( \log_{ x }{ 5 } \right) = 1[/math]I am still stucked. I think need help.