Logarithms - How to Simplify
Logarithms - How to Simplify - Video
Laws of Logarithms
Laws of Logarithms:
Here we have the laws that we are going to use to simplify logarithms either to combine them or to separate them out individually. In these examples we are going to separate them out individually but you can work the other way to combine them using the same laws.
Example 1
Example 1:
We have log(x / [yz]). Now let's separate these different terms individually.
log(x / [y * z])
log(x) - log(y * z)
log(x) - [log(y) + log(z)]
log(x) - log(y) - log(z)
Quotient Rule
Product Rule (Don't forget the extra brackets)
Distributed the negative sign
Our final answer is log(x) - log(y) - log(z).
Be careful with line 3. Since we have a minus sign out in front of log(y * z), we have to add the brackets so we can remember to distribute the minus to each term inside the parenthesis.
Example 2
Example 2:
We have log(x2 y3/ z). Now let's separate these different terms individually.
log(x2 * y3/ z)
log(x2 * y3) - log(z)
log(x2) + log(y3) - log(z)
2log(x) + 3log(y) - log(z)
Quotient Rule
Product Rule (Brackets not needed)
Power Rule
Our final answer is 2log(x) + 3log(y) - log(z).
This time in line 3 we did not need to add brackets since we are simplifying the first term, and we don't have a coefficient term out in front.
Example 3
Example 3:
We have log( 7sqrt(x) ) / 7sqrt(yz). Now let's separate these different terms individually.
log[ 7sqrt(x) / 7sqrt(y * z)]
log{7sqrt(x)} - [ log{7sqrt(y * z)}]
log{7sqrt(x)} - [ log{7sqrt(y)} + log{7sqrt(z)} ]
log{7sqrt(x)} - log{7sqrt(y)} - log{7sqrt(z)}
log{x1/7} - log{y1/7} - log{z1/7}
(1/7) * log{x} - (1/7) * log{y} - (1/7) * log{z}
Quotient Rule
Product Rule (Don't forget the extra brackets)
Distribute the minus sign.
Changed square root to a power.
Power Rule
Our final answer is (1/7) * log{x} - (1/7) * log{y} - (1/7) * log{z}.
Example 4
Example 4:
We have log[4sqrt(x3y4 / z2)]. Now let's separate these different terms individually.
log[4sqrt(x3y4 / z2)]
log[4sqrt(x3y4) - log[4sqrt(z2)]
log[4sqrt(x3)] + log[4sqrt(y4)] - log[4sqrt(z2)]
log[x3/4] + log[y4/4] - log[z2/4]
log[x3/4] + log[y1] - log[z1/2]
(3/4) * log[x] + log[y] - (1/2) * log[z]
Quotient Rule
Product Rule (Extra brackets not needed)
Changed square root to a power.
Simplified.
Power Rule
Our final answer is (3/4) * log[x] + log[y] - (1/2) * log[z].