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Solve for x if log x base5+6 log5 base x=5
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The starting equation is
+ = 5.
Change the base = .
Then the original equation takes the form
+ = 5.
Introduce new variable y = .
Then the equation takes the form
y + = 5.
Multiply all the term by y and reduce to quadratic equation
y^2 - 5y + 6 = 0.
Factor left side
(y-2)*(y-3) = 0.
The roots are y= 2 and y= 3.
From the root y= 2 we get = 2, x = = 25.
From the root y= 3 we get = 3, x = = 125.
At this point, the problem is just solved in full.
ANSWER. The given equation has two solutions, x= 25 and x= 125.
Solved.
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On logarithms and their properties, see these introductory lessons
- WHAT IS the logarithm
- Properties of the logarithm
- Change of Base Formula for logarithms
- Evaluate logarithms without using a calculator
- Simplifying expressions with logarithms
- Solving logarithmic equations
in this site.