Asked by zhamiya moore on Apr 30, 2024
Verified
Solve the logarithmic equation. Round your answer to two decimal places. log10(x−9) +log10x=3\log _ { 10 } ( x - 9 ) + \log _ { 10 } x = 3log10(x−9) +log10x=3
A) x=36.44x = 36.44x=36.44
B) x=495.50x = 495.50x=495.50
C) x=−27.44x = - 27.44x=−27.44
D) x=36.44,x=−27.44x = 36.44 , x = - 27.44x=36.44,x=−27.44
E) there is no solution
Logarithmic Equation
A logarithmic equation is an equation that involves the logarithm of an expression equal to a value.
Decimal Places
The positions to the right of the decimal point in a number, each place representing a power of ten in the base ten number system.
- Work through equations involving logarithms and grasp their crucial role in mathematics.
Verified Answer
CH
Chuck HicksApr 30, 2024
Final Answer :
A
Explanation :
Using the logarithmic property loga(b)+loga(c)=loga(bc)\log_a(b) + \log_a(c) = \log_a(bc)loga(b)+loga(c)=loga(bc) , the equation becomes log10((x−9)x)=3\log_{10}((x - 9)x) = 3log10((x−9)x)=3 . Converting from logarithmic to exponential form gives (x−9)x=103(x - 9)x = 10^3(x−9)x=103 , which simplifies to x2−9x−1000=0x^2 - 9x - 1000 = 0x2−9x−1000=0 . Solving this quadratic equation yields x=36.44x = 36.44x=36.44 as the only valid solution since x=−27.44x = -27.44x=−27.44 would result in a negative argument for the logarithm, which is undefined.
Learning Objectives
- Work through equations involving logarithms and grasp their crucial role in mathematics.