Logarithmic Functions Lesson 5.6
You have used several methods to solve for x when it is contained in an exponent. • You’ve learned that in special cases, it is possible to solve by finding a common base. For example, finding the value of x that makes each of these equations true is straightforward because of your experience with the properties of exponents.
Solving the equation 10x =47 isn’t as straightforward because you may not know how to write 47 as a power of 10. • You can, however, solve this equation by graphing y= 10x and y =47 and finding the intersection—the solution to the system and the solution to 10x 47. Take a minute to verify that 101.672 ≈ 47 is true.
Exponents and Logarithms • In this investigation you’ll discover the connection between exponents on the base 10 and logarithms. • Graph the function f(x)=10x for 1.5≤x≤1.5 on your calculator. Sketch the graph and complete the table of information.
Enter the points for the inverse of f (x) into your calculator and plot them. You will need to adjust the graphing window in order to see these points. Sketch the graph of the inverse function, and complete the table of information about the inverse.
This inverse function is called the logarithm of x, or log(x). Enter the equation y=log(x) into your calculator. Trace your graphs or use tables to find the following values.
Based on your results from the previous step, what is log10x ? Explain. • What is 10log x ? Explain. • Complete the following statements: • If 100 =102, then log 100 = ? . • If 400≈ 102.6021, then log ___ = ____ ? . • If ____=10?, then log 500 ≈_____ ? . • Complete the following statement: If y=10x, then log ____=______ .
The expression log x is another way of expressing x as a power of 10. Ten is the commonly used base for logarithms, so log x is called a common logarithm and is shorthand for writing log10x. You read this as “the logarithm base 10 of x.” Log x is the exponent you put on 10 to get x.
Example A • Solve 4(10x )= 4650.
Example B • Solve 4x=128.
An initial deposit of $500 is invested at 8.5% interest, compounded annually. • How long will it take until the balance grows to $800?