Rules or Laws of Logarithms
Descriptions of Logarithm Rules
The logarithm of the product of numbers is the sum of logarithms of individual numbers.
The logarithm of the quotient of numbers is the difference of the logarithm of individual numbers.
The logarithm of an exponential number is the exponent times the logarithm of the base.
The logarithm of 1 with b > 1 equals zero.
The logarithm of a number that is equal to its base is just 1.
Log of Exponent Rule
The logarithm of an exponential number where its base is the same as the base of the log equals the exponent.
Exponent of Log Rule
Raising the logarithm of a number by its base equals the number.
Example 1: Evaluate the expression using Log Rules log28 + log24 = ?
Express 8 and 4 as exponential numbers with base 2. Then, apply Power Rule followed by Identity Rule. After doing so, you add the resulting values to get your final answer.
So the answer is 5.
Example 2: Evaluate the expression using Log Rules log3162 − log32 = ?
We can’t express 162 as an exponential number with base 3. It appears that we’re stuck since no rules can be applied in direct manner.
However, it’s okay to apply the Logarithm Rules in reverse! Notice that the log expression can be expressed as one or single logarithmic number through the use of the Quotient Rule backwards. Sounds like a plan.
We did it! By applying the rules in reverse, we generated a single log expression that is easily solvable. The final answer here is 4.
Example 3: Evaluate the expression log5500 −2 log52 + log432 + log48 = ?
It looks like there are so many things going on at the same time. First, check if it is possible to simplify each of the logarithmic number. If not, start thinking about some of the logarithmic rules that are obviously applicable.
By observation, we see that there are two bases involved: 5 and 4. So why not put the expressions together having the same base? Let’s simplify them separately.
For log with base 5, apply the Power Rule first followed by Quotient Rule. For log with base 4, apply the Product Rule immediately. Then get the final answer by adding the two values found.
Yep, the final answer is 7.
Example 4: Expand the logarithmic expression log3 (27x2y5).
Inside the parenthesis is a product of factors. Apply the Product Rule to break them up as the sum of individual log expressions. Make sure that you try your best to simplify numerical expressions into exact value whenever possible. Use Rule 5 (Identity rule) as much as you can because it can make the simplification process rather easy.
That’s right! The last line in the detailed solution as shown above is the final answer, although I must admit that they look a bit “unfinished”. As long as we know that we correctly applied the rules, it shouldn’t worry us at all.
Example 5: Expand the logarithmic expression .
The approach is to apply the Quotient Rule first as the difference of two log expressions because they are in fractional form. Then utilize the Product Rule to separate the product of factors as sum of logarithmic expressions.
Example 6: Expand the logarithmic expression .
So this one has a radical expression in the denominator. Remember that the square root symbol is the same as having a power of ½. Express the radical denominator as y½. Just like problem #5, apply the Quotient Rule for logs and then use the Product Rule.
Example 7: Expand the logarithmic expression .
Problem like this may cause you to doubt if indeed you arrived at the correct answer because the final answer can still look “unfinished”. However, as long as you applied the log rules properly in every step, there’s nothing to worry about.
You might notice that we need to apply the Quotient Rule first because the expression is in fractional form.