Category Archives: 6.5 Properties of Logarithms

5.3 Properties of Logarithms, 6.5 Properties of Logarithms

Condense a Logarithm

Problem: Use the properties of logarithms to rewrite the expression as a single logarithm.  Whenever possible, evaluate logarithmic expressions.

ln sqrt{x}- 1/6 ln x +ln root{4}{x}

Soluton:

Logarithmic Expression
ln sqrt{x}- 1/6 ln x +ln root{4}{x}
Use the power rule for logarithms. 
log_b M^p=plog_b M
The coefficient of 1/6 on the middle term becomes the power on the expression inside the logarithm
ln sqrt{x}- 1/6 ln x +ln root{4}{x}
ln sqrt{x}-  ln x^{1/6} +ln root{4}{x}
A radical can be written as a fractional power.  A square root is the same as the one-half power.  A fourth root is the same as the one-fourth power
ln sqrt{x}-  ln x^{1/6} +ln root{4}{x}
ln x^{1/2}-  ln x^{1/6} +ln x^{1/4}
Condense the logarithms using the product and quotient rule.
Product Rule for Logarithms: log_b (MN)=log_b M+log_b N
Quotient Rule for Logarithms: log_b (M/N)=log_b M-log_b N
The expressions inside the logarithm will be positioned in the numerator if the logarithm is positive or will be positioned in the denominator if the logarithm is negative.
ln x^{1/2}-  ln x^{1/6} +ln x^{1/4}
ln {x^{1/2}x^{1/4}}/x^{1/6}
Since these base of the exponential expressions are the same, combine using the power and quotient rules for exponent.
Product Rule for Exponents:
b^Mb^N=b^(M+N)
Quotient Rule for Exponents:
b^M/b^N=b^(M-N)
ln {x^{1/2}x^{1/4}}/x^{1/6}
ln {x^(1/2+1/4)}/x^{1/6}
ln x^(1/2+1/4-1/6)
Find a common denominator to combine the fractions.
ln x^(1/2+1/4-1/6)
ln x^(6/12+3/12-2/12)
ln x^{7/12}

 

Here is a video with a similar example worked out.