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. 

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 .

Using the Change of Base formula we can rewrite the log as follows.

.

Or,

, then find that .

Therefore,

.

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.

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. 

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.