Learning Outcomes
- Solve rational equations by clearing denominators
- Apply the inverse property to simplify logarithms
- Define and use the power rule for logarithms to rewrite expressions
- Write function equations using given conditions
In the First-order Linear Equations section, we will explore how to solve yet another special type of differential equation. Here we will review how to clear fractions from and manipulate rational equations.
Manipulate Rational Equations
Equations that contain rational expressions are called rational equations. For example, [latex] \frac{2y+1}{4}=\frac{x}{3}[/latex] is a rational equation (of two variables).
One of the most straightforward ways to solve a rational equation for the indicated variable is to eliminate denominators with the common denominator and then use properties of equality to isolate the indicated variable.
Solve for [latex]y[/latex] in the equation [latex]\frac{1}{2}y-3=2-\frac{3}{4}x[/latex] by clearing the fractions in the equation first.
Multiply both sides of the equation by [latex]4[/latex], the common denominator of the fractional coefficients.
[latex]\begin{array}{c}\frac{1}{2}y-3=2-\frac{3}{4}x\\ 4\left(\frac{1}{2}y-3\right)=4\left(2-\frac{3}{4}x\right)\\\text{}\,\,\,\,4\left(\frac{1}{2}y\right)-4\left(3\right)=4\left(2\right)+4\left(-\frac{3}{4}x\right)\\2y-12=8-3x\\\underline{+12}\,\,\,\,\,\,\underline{+12}\\ 2y=20-3x\\ \frac{2y}{2}=\frac{20-3x}{2} \\ y=10-\frac{3}{2}x\end{array}[/latex]
Example: Manipulating a Rational Equation
Solve the equation [latex] \frac{x+5}{8}=\frac{7}{y}[/latex] for [latex]y[/latex].
In the next example, we show how to solve a rational equation with a binomial in the denominator of one term. We will use the common denominator to eliminate the denominators from both fractions. Note that the LCD is the product of both denominators because they do not share any common factors.
Example: Manipulating a Rational Equation
Solve the equation [latex] x=\frac{4}{3y+1}[/latex] for [latex]y[/latex].
Apply the Inverse Property to Simplify Logarithms
Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove.
[latex]\begin{array}{l}{\mathrm{log}}_{b}1=0\\{\mathrm{log}}_{b}b=1\end{array}[/latex]
For example, [latex]{\mathrm{log}}_{5}1=0[/latex] since [latex]{5}^{0}=1[/latex] and [latex]{\mathrm{log}}_{5}5=1[/latex] since [latex]{5}^{1}=5[/latex].
Next, we have the inverse property.
[latex]\begin{array}{l}\hfill \\ {\mathrm{log}}_{b}\left({b}^{x}\right)=x\hfill \\ \text{ }{b}^{{\mathrm{log}}_{b}x}=x,x>0\hfill \end{array}[/latex]
For example, to evaluate [latex]\mathrm{log}\left(100\right)[/latex], we can rewrite the logarithm as [latex]{\mathrm{log}}_{10}\left({10}^{2}\right)[/latex] and then apply the inverse property [latex]{\mathrm{log}}_{b}\left({b}^{x}\right)=x[/latex] to get [latex]{\mathrm{log}}_{10}\left({10}^{2}\right)=2[/latex].
To evaluate [latex]{e}^{\mathrm{ln}\left(7\right)}[/latex], we can rewrite the logarithm as [latex]{e}^{{\mathrm{log}}_{e}7}[/latex] and then apply the inverse property [latex]{b}^{{\mathrm{log}}_{b}x}=x[/latex] to get [latex]{e}^{{\mathrm{log}}_{e}7}=7[/latex].
Example: Simplify Logarithms
Simplify each of the following.
a) [latex]e^{\ln(7x)}[/latex]
b) [latex]\log_9(9^4)[/latex]
c) [latex]5^{\log_5(3y)}[/latex]
Try It
Simplify each of the following.
a)[latex]b^{\log_b(15x)}[/latex]
b) [latex]\log_2(2^a)[/latex]
c) [latex]e^{\ln(4y)}[/latex]
Use the Power Property for Logarithms
A General Note: The Power Rule for Logarithms
The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.
[latex]{\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}M[/latex]
Example: Expanding a Logarithm with Powers
Rewrite [latex]{\mathrm{log}}_{2}{x}^{5}[/latex].
Try It
Rewrite [latex]\mathrm{ln}{x}^{2}[/latex].
Example: Rewriting an Expression as a Power before Using the Power Rule
Rewrite [latex]{\mathrm{log}}_{3}\left(25\right)[/latex] using the power rule for logs.
Try It
Rewrite [latex]\mathrm{ln}\left(\frac{1}{{x}^{2}}\right)[/latex].
Write Function Equations Using Given Conditions
(See Module 4, Skills Review for Basics of Differential Equations)
