Solving an equation with a logarithm in the

Aryan Salinas

Aryan Salinas

Answered question

2022-04-01

Solving an equation with a logarithm in the exponent
I try to solve the following equation:
(N+1)logN125=216
I know the answer is 5 here but how could I rewrite the equations so I can solve it?
I tried to take the log of both sides but that didn't help me because I got stuck. Could anyone please explain me how to do this?
Thanks!

Answer & Explanation

Harry Gibson

Harry Gibson

Beginner2022-04-02Added 13 answers

Hint: Use the rules of logarithm, especially the power rule and change of base rule.
Take log6 on both sides, and then simplify the equation to obtain
log65=logN+1N
Observe that the graph of logN+1N is monotonic (for example, by differentiating), hence the unique answer is N=5.
kaosimqu5t

kaosimqu5t

Beginner2022-04-03Added 10 answers

Note that 216=63, and 125=53, so use the logarithm power rule:
log(ab)=bloga to write:
(N+1)logN53=(N+1)3logN5=63
That is
[(N+1)3]logN{5}=(5+1)3
and your solution is apparent.

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