Category Archives: 5.3 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.

5.3 Properties of Logarithms, 5.4 Exponential and Logarithmic Equations, 6.6 Logarithmic and Exponential Equations, Logarithmic Equations

Solve the Logarithmic Equation by the one to one property

Example:

2 log_3(7-x)-log_3 2=log_3 18

Solution:

 

The logarithmic equation
2 log_3(7-x)-log_3 2=log_3 18
Use the power rule and the quotient rule to condense to a single logarithm
2 log_3(7-x)-log_3 2=log_3 18
log_3(7-x)^2-log_3 2=log_3 18
log_3((7-x)^2/ 2)=log_3 18
Since both sides of the equation have the same log base the expressions inside the logarithms must be equal
log_3((7-x)^2/ 2)=log_3 18
(7-x)^2/ 2= 18
Clear the denominator by multiplying by 2 on both sides and simplifying
(7-x)^2/ 2= 18
2*(7-x)^2/ 2= 2*18
(7-x)^2= 36
Get rid of the square by square rooting both sides and simplifying
(7-x)^2= 36
sqrt{(7-x)^2}= sqrt{36}
7-x= pm 6
Get x by itself by subtracting 7 on both sides
7-x= pm 6
7-7-x=-7 pm 6
-x=-7 pm 6
Get x by itself by dividing both sides by negative 1
-x=-7 pm 6
-x/-1={-7 pm 6}/-1
x=7 pm 6
x=7 + 6 or x=7 - 6
x=13 or x=1
Check x=13
2 log_3(7-13)-log_3 2=log_3 18
2 log_3(-6)-log_3 2=log_3 18
Log of a negative is undefined.  Exclude this solution.
Check x=1
2 log_3(7-1)-log_3 2=log_3 18
2 log_3(6)-log_3 2=log_3 18
log_3(6)^2-log_3 2=log_3 18
log_3 36-log_3 2=log_3 18
log_3 36/2=log_3 18
log_3 18=log_3 18
Keep this solution.

The solution to the equation is x=1.

Here is a youtube video that is similar.