Logarithmic Functions - Pre-Calculus

The problem asks us to evaluate the following logarithm:

According to the definition of a logarithm, what this equation is asking us is "5 to what power equals 7?":

Using the properties of logarithms, if we take the log of both sides we get the following equation and simplification:

In order to evaluate the logarithmic expression, we have to remember the notation of a logarithm and what it means:

Any logarithm expressed in the form above is simply asking "a to what power equals b?" So we're trying to find what power the base must be raised to in order to obtain the value in parentheses. If we look at our expression in particular:

The first term is asking "5 to what power equals 1/25?" while the second is asking "7 to what power equals 49?" Setting these questions up mathematically, we can find the values of each logarithm:

So the value of the first term is -2, and the value of the second term is 2, which gives us:

The notation of this logarithm is asking "5 to what power equals 125?" If we set this up mathematically, we can simplify either side until it is readily apparent what the value of x is, which is the value of the logarithm:

So to answer our question, 5 to the power of 3 is 125, so the answer is 3.

To evaluate, you must convert 256 to a power of 2. Then the log and 2 will cancel to leave you with your answer.

The function is undefined unless . Thus is undefined unless because the function has been shifted left.

To find the range of this particular function we need to first identify the domain. Since we know that is a bound on our function.

From here we want to find the function value as approaches .

To find this approximate value we will plug in into our original function.

This is our lowest value we will obtain. As we plug in large values we get large function values.

Therefore our range is:

When evaluating a log you are asking what power your base must be raised by to get your argument.

If you cannot directly figure that out you can use a change of base formula instead.

For , is the exponent of base 5 and is the product. Therefore, when , and when , . As a result, the correct graph will have values of 5 and 125 at and , respectively.

Looking at the diagram, we can see that when , . Since represents the exponent and represents the product, and any base with an exponent of 1 equals the base, we can determine the base to be 0.5.

For this function, represents the exponent and y represents the product of the base 2 and its exponent. On the diagram, it is clear that as the value increases, the value increases exponentially and at , . Those two characteristics of the graph indicate that x is the exponent value and the base is equal to 2.

Looking at the graph, the y-value diminishes exponentially as decreases and increases rapidly as the x-value increases, which indicates that is the exponent value for the equation.

Also, when and when , which can be expressed as and , respectively.

This indicates that the diagram is consistent with the function .

It is important to remember a logarithm is really just an exponent. In fact when you see remember that the expression is only asking what exponent must "a" be raised to in order to obtain "x"!

Now that we have substituted the value of x reread the problem by the rule above. What power must 3 be raised to in order to obtain 243?

5 is the power to which 3 must be raised to obtain 243.

Given the property

and because

we obtain that

Using a calculator, evaluate

To evaluate, use a calculator to find

To solve, simply find the matching base and use properties of logs to simplify.

Some properties of logs include the following:

Thus,

To solve, remember the following rules for logarithms.

Thus,

Remember, if a base isn't specified, it is 10.