Dealing with Logarithms. log(b^x+a)=log(c) What methods/techniques are available to solve for x in the following type of situation: log(b^x+a)=log(c) The only log methods I have been exposed to are using the power laws and bring x out, which you cannot do in this case. Thanks for your help.

Howard Nelson

Howard Nelson

Answered question

2022-11-18

Dealing with Logarithms. log ( b x + a ) = log ( c )
What methods/techniques are available to solve for x in the following type of situation:
log ( b x + a ) = log ( c )
The only log methods I have been exposed to are using the power laws and bring x out, which you cannot do in this case.
Thanks for your help.

Answer & Explanation

Zoe Andersen

Zoe Andersen

Beginner2022-11-19Added 16 answers

You can use the function e x as inverse of log:
Apply e x on both sides:
b x + a = c
Minus c:
b x = c a
Apply log and use exponent law:
x log ( b ) = log ( c a )
Divide:
x = log ( c a ) log ( b )
Keep in mind that log is only defined on stric positive values, and you should not divide by 0 as well, that means we assume b 1
Clara Dennis

Clara Dennis

Beginner2022-11-20Added 5 answers

The log function is bijective, so if log ( b x + a ) = log ( c ), then b x + a = c. Don't forget that b x + a should be strictly positive.
b x = c a if x = b log ( c a ). This is exactly the definition of b log ( c a ): to which power should we raise b in order to get c a

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