3.5 Properties and applications of logarithmic functions

Introduction

Key properties of logarithms

The graph of the natural logarithm

In the following questions, we investigate how $\log_{10}(a \cdot b)$ can be equivalently written in terms of $\log_{10}(a)$ and $\log_{10}(b)$.

Write $10^x \cdot 10^y$ as $10$ raised to a single power. That is, complete the equation

by filling in the box with an appropriate expression involving $x$ and $y$.

Solve the equation $7 \cdot 3^t - 1 = 5$ exactly for $t$.

$3^t = 5$

$4 \cdot 2^t - 2 = 3$

$3.7 \cdot (0.9)^{0.3t} + 1.5 = 2.1$

$72 - 30(0.7)^{0.05t} = 60$

$\ln(t) = -2$

$3 + 2\log_{10}(t) = 3.5$

Let $f(t) = 1 - e^{-(t-1)}$ and $g(t) = \ln(t)$. Plot each function on the same set of coordinate axes. What properties do the two functions have in common? For what properties do the two functions differ? Consider each function's domain, range, $t$-intercept, $y$-intercept, increasing/decreasing behavior, concavity, and long-term behavior.

Let $h(t) = a - be^{-k(t-c)}$, where $a$, $b$, $c$, and $k$ are positive constants. Describe $h$ as a transformation of the function $E(t) = e^t$.

Let $r(t) = a + b\ln(t-c)$, where $a$, $b$, and $c$ are positive constants. Describe $r$ as a transformation of the function $L(t) = \ln(t)$.

Data for the height of a tree is given in the following table; time $t$ is measured in years and height is given in feet. At http://gvsu.edu/s/0yy, you can find a Desmos worksheet with this data already input.

$t$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11

$h(t)$ | 6 | 9.5 | 13 | 15 | 16.5 | 17.5 | 18.5 | 19 | 19.5 | 19.7 | 19.8

Do you think this data is better modeled by a logarithmic function of form $p(t) = a + b\ln(t-c)$ or by an exponential function of form $q(t) = m + ne^{-rt}$. Provide reasons based in how the data appears and how you think a tree grows, as well as by experimenting with sliders appropriately in Desmos. (Note: you may need to adjust the upper and lower bounds of several of the sliders in order to match the data well.)