First, simplify  to 

Then, from the information in the graph we observe that we should find the  intercept, the point at which 

so,  and thus A is the curve we are looking for since it exclusively intersects the -axis at 

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:

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 ,  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: