Simplifying Logarithms - Math

The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.

If we encounter two logarithms with the same base, we can likely combine them. In this case, we can use the reverse of the above identity.

The rule for the addition of logarithms is as follows:

.

As an application of this,.

Using properties of logs we get:

Recall the log rule:

In this particular case, and . Thus, our answer is .

Since the bases of the logs are the same and the logarithms are added, the arguments can be multiplied together. We then simplify the right side of the equation:

The logarithm can be converted to exponential form:

Factor the equation:

Although there are two solutions to the equation, logarithms cannot be negative. Therefore, the only real solution is .

Using the logarithm rules, exponents within logarithms can be removed and simply multiplied by the remaining logarithm. This expression can be simplified as

Use the power reducing theorem:

and

According to the rule for exponents of logarithms,. As a direct application of this,.

Based on the definition of exponents, .

Then, we use the following rule of logarithms:

Thus, .

By logarithmic properties:

;

Combining these three terms gives the correct answer:

Remembering the rules for logarithms, we know that .

This tells us that .

This becomes , which is .

Use the change of base formula to solve this equation.

Recall that log implies base if not indicated.Then, we break up . Thus, we have .

Our log rules indicate that

.

So we are really interested in,

.

Since we are interested in log base , we can solve without a calculator.

We know that , and thus by the definition of log we have that .

Therefore, we have .

Use the rule

In order to solve this problem you must understand the product property of logarithms and the power property of logarithms . Note that these apply to logs of all bases not just base 10.

log of multiple terms is the log of each individual one:

now use the power property to move the exponent over:

We can rewrite the terms of the inner quantity. Change the negative exponent into a fraction.

This means that:

Split up these logarithms by addition.

According to the log rules, the powers can be transferred in front of the logs as coefficients.

The answer is:

The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.

If we encounter two logarithms with the same base, we can likely combine them. In this case, we can use the reverse of the above identity.

The rule for the addition of logarithms is as follows:

.

As an application of this,.

Using properties of logs we get:

Recall the log rule:

In this particular case, and . Thus, our answer is .