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Find y in terms of x: log y - 3 log x = 3
Mathematics
Grade 10 (Junior High)
Question Content
If log y - 3 log x = 3, find an expression for y in terms of x.
Correct Answer
y = 1000x^3
Detailed Solution Steps
1
Step 1: Use the power rule of logarithms to rewrite 3 log x as log(x^3).
2
Step 2: Rewrite the equation: log y - log(x^3) = 3.
3
Step 3: Apply the quotient rule of logarithms: log(y/x^3) = 3.
4
Step 4: Convert the logarithmic equation to exponential form (base 10, since no base is specified): y/x^3 = 10^3 = 1000.
5
Step 5: Solve for y: y = 1000x^3.
Knowledge Points Involved
1
Power Rule of Logarithms
k log_a m = log_a(m^k) for any real number k, positive m, and a>0, a≠1. Used to move coefficients of logarithms as 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
Logarithm to Exponential Conversion
If log_a b = c, then a^c = b (for a>0, a≠1, b>0). Converts logarithmic equations to exponential form for easier solving.
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