Logarithms - Math

For any equation , . Thus, we are trying to determine what power of 10 is 1000. , so our answer is 3.

is equivalent to . In this case, you know the value of (the argument of a logarithmic equation) and b (the answer to the logarithmic equation). You must find a solution for the base.

Most of us don't know what the exponent would be if and unfortunately there is no on a graphing calculator -- only (which stands for ).

Fortunately we can use the base change rule:

Plug in our given values.

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, .

First, change 25 to so that both sides have the same base. Once they have the same base, you can apply log to both sides so that you can set their exponents equal to each other, which yields .

Change 49 to so that both sides have the same base so that you can apply log. Then, you can set the exponential expressions equal to each other .

Thus,

Change 81 to so that both sides have the same base. Once you have the same base, apply log to both sides so that you can set the exponential expressions equal to each other (). Thus, .

Change the right side to so that both sides have the same bsae of 10. Apply log and then set the exponential expressions equal to each other

Change 64 to so that both sides have the same base. Apply log to both sides so that you can set the exponential expressions equal to each other

.

Thus, .

Change the left side to and the right side to so that both sides have the same base. Apply log and then set the exponential expressions equal to each other (). Thus, .

Change 125 to so that both sides have the same base. Apply log and then set the exponential expressions equal to each other so that . Upon trying to isolate , it becomes clear that there is no solution.

Change the right side to so that both sides are the same. Apply log to both sides so that you can set the exponential expressions equal to each other ().