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Section 0.3 Review of Exponential Functions and Logarithms
In MAT 103, you learned how to model exponential situations with the equation
\begin{equation*}
f(t)=ab^t
\end{equation*}
where \(a\) is a nonzero real number and \(b\) is a positive real number.
You also learned how logarithmic functions can help us solve for an unknown exponent in an exponential equation. If \(x\) is a positive number then \(\log_{b}(x)\) is the exponent of \(b\) that gives \(x\text{.}\) That is
\begin{equation*}
y=\log_{b}(x) \text{ if and only if } b^{y}=x
\end{equation*}
The number \(b\) is called the base of the logarithm and is always larger than 0.
Lastly, you learned how to apply exponent rules to simplify expressions involving exponents. Below is a list of exponent rules where
\(a \neq 0\text{,}\) \(b \neq 0\text{,}\) and
\(m\) and
\(n\) are real numbers.
\(\displaystyle a^n \cdot a^m=a^{m+n}\)
\(\displaystyle \frac{a^m}{a^n}=a^{m-n}\)
\(\displaystyle \left(a^m \right)^n=a^{mn}\)
\(\displaystyle \left(ab \right)^m=a^m \cdot b^m\)
\(\displaystyle a^{-n}=\frac{1}{a^n}\)
We can expand our use of exponents to include radicals by representing
\(\sqrt[n]{a}\) as
\(a^{\frac{1}{n}}\) and
\(\sqrt[n]{a^m}\) as
\(a^{\frac{m}{n}}\text{,}\) where
\(n>0\) and
\(m\) is a real number. We can then apply the rules listed above.
Problem 0.3.1 .
Let
\(f\) be defined by the table below.
Table 0.3.2.
\(f(x)\) 6
3
1.5
0.75
0.375
What type of function would you use to model the relationship in function
\(f\text{?}\) Why?
Write an exponential function for
\(f\text{.}\)
Evaluate
\(f(3)\text{.}\)
Use Desmos to solve
\(f(x)=11\text{.}\) Sketch a graph and label your solution.
Use logarithms to solve
\(f(x)=1\text{.}\)
What are the domain and range of
\(f\text{?}\)
Write an equation for the inverse of
\(f\text{,}\) \(f^{-1}\text{.}\)
What are the domain and range of
\(f^{-1}\text{?}\)
Problem 0.3.3 .
The half-life of iodine-135 is 8 days.
If you begin with a 100 mg sample, write an equation for
\(M(d)\) that models the amount of iodine-135 remaining in the sample after
\(d\) days.
What will the mass of the sample be after 20 days?
How long will it take for the sample to decay to a mass of 70 mg?
Problem 0.3.4 .
Since 1910 human population has been growing exponentially. In 1910, the world’s population was 1.75 billion and in 2017 it was 7.53 billion.
Write an equation for
\(P(t)\) that models human population
\(t\) years after 1910.
Use your equation to find the world’s population in 2010.
The world’s population was known to be 6.933 billion in 2010. How close is this to your predicted population from part
Item 2 ?
Use your model to predict how long it take for world’s population to reach 8 billion?
Problem 0.3.5 .
Simplify each expression. Write your answer without negative exponents.
\(\displaystyle x(x^6)(x^{-7})\)
\(\displaystyle \displaystyle{\frac{-20a^5b}{50a^{-1}b^3}}\)
\(\displaystyle \displaystyle{\left( \frac{x^3y^{-3}}{x^{-2}} \right)^{-1}}\)
\(\displaystyle -7c^4(-2c^{-2})^3\)
Problem 0.3.6 .
Write each expression as a power of
\(x\text{.}\)
\(\displaystyle (\sqrt[5]{x})^6\)
\(\displaystyle \displaystyle{\frac{1}{x\sqrt{x}}}\)
\(\displaystyle \sqrt[4]{\sqrt[3]{x}}\)
\(\displaystyle \displaystyle{\left(\frac{x}{x^3}\right)^{-\frac{2}{3}}}\)
Problem 0.3.7 .
Write each expression as a sum of powers.
\(\displaystyle \sqrt{x}(x^2+2x+\sqrt{x}+\frac{1}{\sqrt{x}})\)
\(\displaystyle \displaystyle{\frac{x^2-3x-2}{x^2}}\)
