4.4 Exponential and Logarithmic Equations
Example using the One-to-One Property of Logarithms to Solve Logarithmic Equations
A graphing utility's TABLE feature can be used to verify that {3} is the solution set of the logarithmic equation: y1= ln(x+2) - ln(4x+3) y2= ln (1/x) y1 and y2 are equal when x=3.
What is a logarithmic equation?
A logarithmic equation is an equation containing a variable in a logarithmic expression. See the image for examples of logarithmic equations.
Example: Solving an Exponential Equation that is Quadratic in Form (using u=e^x)
If you graph the equation, you'll see the two x-intercepts are 0 and approximately 1.10, which verifies the algebraic solution.
Why and how do you Condense a Logarithmic Equation?
It is sometimes necessary to use properties of logarithms to condense logarithms into a single logarithm so you can rewrite it in exponential form to solve for the variable. The example uses the product rule for logarithms to obtain a single logarithmic expression on the left side.
How do you Solve a Logarithmic Equation?
1. Express the equation in the form logbM=c. 2. Use the definition of a logarithm to rewrite the equation in exponential form: logbM=C means b^c=M *logarithms are exponents* 3. Solve for the variable. 4. Check proposed solutions in the original equation. *Include in the solution set only values for which M > 0. Logarithmic expressions are defined only for logarithms of positive real numbers. Always check proposed solutions of a logarithmic equation in the original equation. Exclude from the solution set any proposed solution that produces the logarithm of a negative number or the logarithm of 0. A negative number can belong to the solution set of a logarithmic equation as long as it does not produce the logarithm of a negative number.
How do you use the One-to-One Property of Logarithms to Solve Logarithmic Equations?
1. Express the equation in the form logbM=logbN. *this form involves a single logarithm whose coefficient is 1 on each side of the equation. 2. Use the one-to-one property to rewrite the equation without logarithms: If logbM=logbN, then M=N. 3. Solve for the variable. 4. Check proposed solutions in the original equation. *Include in the solution set only values for which M > 0 and N > 0.
What is an exponential equation and how do you solve it?
An exponential equation is an equation containing a variable in an exponent. If b^M = b^N then M=N 1.Rewrite the equation in the form b^M = b^N. *write each side as a power of the same base* 2. Set M=N. *Multiply the exponents if the exponential expression is raised to a power* 3. Solve for the variable
How do you solve exponential equations that cannot be rewritten so that each side has the same base?
Use logarithms. 1. Isolate the exponential expression on one side. 2.Take the logarithm on both sides; take the common logarithm,10, for base 10 and the natural logarithm,e, for bases other than 10. 3. Simplify using one of the following properties: ln b^x = x ln b or ln e^x = x or log 10^x = x. 4. Solve for the variable.
Example: Solving an Exponential Equation
Verify the solution set by graphing both sides of the equation. y^1 = 2^3x-8 y^2 = 16 You can see that the graphs have an intersection point whose x-coordinate is 4, thus verifying that {4} is a solution set of the given exponential equation.
Can a negative number belong to the solution set of a logarithmic equation?
Yes. In the example the solution set is {-12}. Although -12 is negative, it does not produce the logarithm of a negative number in log2(x+20)=3, the given equation. Note that the domain of the expression log2(x+20) is (-20,infinity), which includes negative numbers such as -12.
