High School: Functions
Linear, Quadratic, and Exponential Models HSF-LE.A.4
4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Students should be familiar with the conversion of an exponential function into logarithmic form. All this does is translate to a simple conversion formula that you need to teach them.
If we break it down to its simplest form, we can see that an exponential equation can be written as by = x, where b, y, and x are just numbers. We can then say that our exponential model expressed as logarithm look like: logbx = y.
Students should know that the two equations represent the same exact relationship. Make sure to stress that logarithms are simply another way to rewrite exponents.
In the world of logarithms, the most commonly used bases (the b in logbx = y) are either 2, 10, or e. Students should also know to equate the term "natural log" or ln with loge, and to assume a base of 10 when no b is given.
Our equation of 32 = 9, in logarithmic form, is log3(9) = 2. To calculate this out on a graphing calculator (since they typically only have a log with base 10 or e) we would equate logbx to . In this case it's: = 2, which confirms both our logarithmic and exponential equation.
If we're given 102 = 100, we could rearrange it to be log10(100) = 2, or simply log(100) = 2. If we wanted to double check, we could calculate ln(100)/ln(10) into our calculator, and it should yield the same result.
Students should also be able to make simple cancellations when working these problems out algebraically. For example, ln e = 1 because
(Or because, according to our definition, e should have an exponent of 1 to make it equal itself). They should also be able to manipulate logarithmic expressions to help reduce logarithms to their simplest form.
Once students have gotten their logarithms to the most simplified form, they can use technology (whether that means graphing calculators, abacuses, or the mighty Google machine) to help solve the logarithm.