Asked by tt
Solve: log4x+log3=log84
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Answered by
GPT-5 mini
AI
Interpreting the equation as log_4 x + log_3 x = log_8 4 (so both logs on the left have argument x):
Compute the right-hand side:
log_8 4 = ln 4 / ln 8 = (2 ln 2)/(3 ln 2) = 2/3.
Write the left-hand side with natural logs:
ln x (1/(2 ln 2) + 1/ln 3) = 2/3.
So
ln x = (2/3) / (1/(2 ln 2) + 1/ln 3)
= (4/3) * (ln 2 · ln 3) / ln 12.
Therefore
x = exp[(4 ln 2 · ln 3)/(3 ln 12)] ≈ 1.5057.
(That value satisfies x>0 as required.)
Compute the right-hand side:
log_8 4 = ln 4 / ln 8 = (2 ln 2)/(3 ln 2) = 2/3.
Write the left-hand side with natural logs:
ln x (1/(2 ln 2) + 1/ln 3) = 2/3.
So
ln x = (2/3) / (1/(2 ln 2) + 1/ln 3)
= (4/3) * (ln 2 · ln 3) / ln 12.
Therefore
x = exp[(4 ln 2 · ln 3)/(3 ln 12)] ≈ 1.5057.
(That value satisfies x>0 as required.)
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