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Solve Exponential Equation: 3^(1.5x - 4) = 8
Mathematics
Grade 10 (Junior High)
Question Content
Solve the equation: 3^(1.5x - 4) = 8
Correct Answer
x = (4 + log3 8)/1.5 = (4 + 3log3 2)/1.5 ≈ 3.77
Detailed Solution Steps
1
Step 1: Take the logarithm (base 3) of both sides: log3(3^(1.5x-4)) = log3 8.
2
Step 2: Use the inverse property log_a(a^b)=b: 1.5x - 4 = log3 8.
3
Step 3: Solve for x: 1.5x = 4 + log3 8 → x = (4 + log3 8)/1.5.
4
Step 4: Optional: Rewrite log3 8 as 3log3 2, or convert to base 10: log3 8 = ln8/ln3 ≈ 1.8928, so x≈(4+1.8928)/1.5≈3.77.
Knowledge Points Involved
1
Inverse Property of Logarithms
log_a(a^b) = b for any real b, a>0, a≠1. Used to simplify logarithms of exponential terms with the same base.
2
Change of Base Formula
log_a b = log_c b / log_c a for any positive c≠1. Converts logarithms between bases to calculate decimal approximations.
3
Solving Exponential Equations
To solve a^f(x)=b, take the logarithm of both sides, use logarithm properties to isolate the variable x.
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