How do you solve logarithmic equations like this one? How do you solve 3 log &#x2061;<!

dream13rxs 2022-06-29 Answered
How do you solve logarithmic equations like this one?
How do you solve
3 log ( x 15 ) = ( 1 4 ) x ?
The solution is approximately 16. How would you solve a logarithmic equation with an solution approximately equal to a number without using a graphing calculator?
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Answers (2)

engaliar0l
Answered 2022-06-30 Author has 13 answers
You don't say what base of logs you are using. The approach will be the same in any case-I will assume natural logs. We must have x > 15 or the logarithm is not defined. In that case, the right side will be very small and positive. We need x > 16 to make the left side positive. Define a = x 16, where we expect a to be very small, so we will use the first term of the Taylor series.
3 log ( x 15 ) = ( 1 4 ) x 3 log ( 1 + a ) = ( 1 4 ) 16 + a 3 4 16 a = 4 a
This shows a 1 3 4 16 with both sides very close to 1, so x 16 + 1 3 4 16

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Jameson Lucero
Answered 2022-07-01 Author has 5 answers
The point is that for x > 10 the right hand side is smaller than 1 / 1000000 , so you are essentially solving for L H S = 0. There is no general method if you are not so lucky.

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