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Solve Logarithmic Equation: log8x + 2log(3x) - log4x = log(6-12x)
Mathematics
Grade 10 (Junior High)
Question Content
Solve the equation: log 8x + 2 log(3x) - log 4x = log(6 - 12x)
Correct Answer
x = 1/3
Detailed Solution Steps
1
Step 1: Apply power rule to 2 log(3x): log((3x)^2)=log(9x²).
2
Step 2: Apply product rule to log8x + log9x²: log(8x*9x²)=log(72x³).
3
Step 3: Apply quotient rule to left side: log(72x³/(4x))=log(18x²).
4
Step 4: Set arguments equal: 18x² = 6 -12x.
5
Step 5: Rearrange to standard quadratic form: 18x²+12x-6=0 → 3x²+2x-1=0.
6
Step 6: Factor: (3x-1)(x+1)=0. Solutions x=1/3 or x=-1.
7
Step 7: Check validity: x=-1 makes log8x and log(6-12x) undefined (negative arguments). x=1/3 is valid (all arguments positive).
Knowledge Points Involved
1
Product Rule of Logarithms
log_a m + log_a n = log_a(mn) for positive m,n, a>0, a≠1. Combines added logarithms into a single logarithm of the product.
2
Quadratic Equations from Logarithmic Equations
Simplifying logarithmic equations often results in quadratic equations, which can be solved by factoring, completing the square, or quadratic formula.
3
Validity of Logarithm Arguments
All arguments of logarithm functions in the original equation must be positive. Discard any solutions that result in negative or zero arguments.
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