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Solve Logarithmic Equation: 5 log x - 3 log(2x) = log 7x
Mathematics
Grade 10 (Junior High)
Question Content
Solve the equation: 5 log x - 3 log(2x) = log 7x
Correct Answer
x = (7*8)/1 = 56
Detailed Solution Steps
1
Step 1: Apply the power rule to each term: log(x^5) - log((2x)^3) = log(7x).
2
Step 2: Simplify (2x)^3 = 8x^3, so the equation becomes log(x^5) - log(8x^3) = log(7x).
3
Step 3: Apply the quotient rule to the left side: log(x^5/(8x^3)) = log(x²/8).
4
Step 4: Set arguments equal (one-to-one property): x²/8 = 7x.
5
Step 5: Rearrange: x² = 56x → x² -56x=0 → x(x-56)=0. Solutions x=0 or x=56.
6
Step 6: Check validity: x=0 makes logx undefined, so discard. x=56 is valid (all log arguments are positive).
Knowledge Points Involved
1
Power Rule of Logarithms
k log_a m = log_a(m^k) for real k, positive m, a>0, a≠1. Converts coefficients of logarithms to exponents inside the logarithm.
2
Quotient Rule of Logarithms
log_a m - log_a n = log_a(m/n). Combines two subtracted logarithms into a single logarithm of the quotient.
3
Validity of Logarithm Inputs
Logarithm functions are only defined for positive arguments. All solutions must be checked to ensure all log terms in the original equation are defined.
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