\text{Condense the following expression into a single logarithm using the pro

Michael Dennis

Michael Dennis

Answered question

2021-11-11

Condense the following expression into a single logarithm using the properties of logarithms. Assume x>0 and  x6
log5(x28x+12)2log5(x6)

Answer & Explanation

pseudoenergy34

pseudoenergy34

Beginner2021-11-12Added 22 answers

Step 1
According to the definition of logarithm, if y=ax,  then  loga(y)=x, where a is called the base of the logarithm. According to the base changing rule, logarithm of any number to the base a can be converted into another, that is loga(y)=logb(y)logb(a)
If the base of the logarithm is 10, then it is called the common logarithm, example log10(y),log10(102) etc. Similarly if the base of the logarithm is e, then it is called natural logarithm, examples loge(y),loge(102.25) etc.
Step 2
To express log5(x28x+12)2log5(x6) as a single logarithm consider the following steps.
According to the power property of logarithm, the logarithm of the exponent of a number (pq) is equal to the exponent times the logarithm of the number (p), that is, qloga(p)=loga(pq). Use power property to express 2log5(x6)  as  log5(x6)2
According to the quotient property of logarithm, the logarithm of the quotient of two numbers is equal to the difference of their individual logarithms, that is, loga(m)loga(n)=loga(mn). In this problem a=5,m=x28x+12  and  n=(x6)2
The expression x28x+12 can be written as (x6)(x2). To simplify the quantity inside the parenthesis of log5[(x28x+12)(x6)2], cancel out the common x−6 factor.
log5(x28x+12)2log5(x6)=log5(x28x+12)log5(x6)2
=log5[(x28x+12)(x6)2]
=log5[(x6)(x2)(x6)2]
=log5[(x2)(x6)

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