Once you are familiar with logarithms and exponential functions, you can look at logarithmic functions. Logarithms are basically another way of writing exponents and logarithmic functions are inverses of exponential functions.
An important definition that we should keep in mind is the definition of logs, because it will help tremendously when dealing with logarithmic functions and various types of log problems.
To reiterate, a logarithm is another way of writing an exponent. This definition works in both directions (converting from exponential form to logarithmic and back). The domain for logarithmic functions will be all positive real numbers for x and the range will be all real numbers for y.
When we write log(x) without a base, it is understood that the base is 10. On a calculutor, there are two types of logs, a base ten log and the natural log ln(x). The natural log is of base e, which we found is a unique exponential function. Many problems will deal with e and we will have to use the natural log ln to evaluate and graph the function.
Graphing the exponential function and natural log function, we can see that they are inverses of each other.
Let's graph the function f(x) = log(x+2) of base 4.
We can use the definition of logs to rewrite this in exponential form. We can see that the base is 4, the exponent is y, and the log will set to be (x+2).
We can then plug in values for y and yield our x values. Though we usually plug in x values to find our y values, it is much easier in this form to plug in various values for y.
Notice that now we have a vertical asymptote at x = -2, and a point at (-1,0). This is exactly the opposite of exponential functions, which have horizontal asymptotes and a point at (0,1). All logarithmic functions will have a vertical asymptote and pass through the point at distance 1 from the vertical asymptote in the direction it opens up. This point will always be the x intercept. To find it, we can set y equal to 0 and solve for x.
Let's look at the graph of f(x) = ln(5-x).
Notice that our domain is x < 5. If our input is greater than 5, our output will not be defined. The x intercept is given as
In the first step, we can use the definition of logs to rewrite the equation and solve for x. Our x intercept is then (4,0).
Inverse Properties of Logarithms
Inverse Property I
This means that whenever the base of the log matches the base of the inside log, the log will equal the exponent of the inside base. This is only if the bases match.
Inverse Property II
This means whenever the base raised to a log has the same base, then it simplifies to whatever is inside the log. This is confirming that logs are another way of writing exponents, just like subtracting is another way of writing addition and division is another way of writing multiplication. Again, this property only works if the the base b is the same as the base b of the log.
Let's evaluate some logarithmic equations and expressions to practice our knowledge of properties of logs.
Solve for x
Solve for x
Write as one logarithm
Solve for x