MATH 133 – Calculus with Fundamentals 1 Logarithm Functions September 21, 2015

Background

As my high school math teacher Mr. Brennan said, “a logarithm is an exponent.” He was a total nerd, but he was right! Of course, this is another way of saying the key property defining the log_{b} function:

y = log_{b} x ⟺ x = b^{y},

so log_{b}(b^{y}) = y for all real y, and b^{log_{b}(x)} = x for all x > 0. The most important logarithm function for us is the natural logarithm ln(x), which is the same as log_{e}(x) where e ≐ 2.71828 ⋯ is (for now) a somewhat mysterious number(!) From the properties of exponents, (see the table on page 41 of our text), we get the key properties of the ln function:

(1) ln(x_{1}x_{2}) = ln(x_{1}) + ln(x_{2}),

(2) ln (\frac{x_{1}}{x_{2}} ) = ln(x_{1}) − ln(x_{2}), and

(3) ln (A^{B}) = B ln(A),

and similarly for ln_{b} for any b > 0. Today, we want to practice using these properties of the logarithm functions to simplify expressions and solve equations.

Questions

1) Simplify and compute the value. (Do these without using a calculator!)

(a) log_{2} (\frac{1}{128})

(b) log_{5}(125)

(c) ln(e^{4}) + ln (\frac{1}{e^{5}} )

2) Solve each of these equations for the unknown by using properties of logarithms. Get an exact answer (expressed using logarithms), then find a decimal approximation using a calculator.

(a) e^{5x+1} = 2.

(b) 2^{2x} = 3^{3x−5}.

(c) e^{2t+1} = 7^{t−4}

3) (a) Explain why f(x) = e^{2(x−1)} + 1 has an inverse function. (Hint: What is the property that says a function has an inverse function? Sketch the graph y = e^{2(x−1)} + 1 to see that this function “has it.”)

(b) Find a formula for f^{−1} for the function from part (a). Sketch the graph y = f^{−1}(x) on the same axes as you had in part (a).