5 6 Solving Exponential and Logarithmic Equations Objectives

5. 6 Solving Exponential and Logarithmic Equations Objectives: 1. Solve exponential and logarithmic equations. 2. Solve a variety of application problems by using exponential and logarithmic equations.

Methods of Solving Exponential Equations 1. Rewrite the bases of the powers on both sides so they are the same. This method is the fastest to use but only works for certain situations. It uses a fact that we often take for granted. For instance, in the equation 2 x = 23, we would say that x = 3. If the bases are equal, then the exponents must be equal as well. If the equation were modified slightly, say 2 x – 1 = 23, we would still set the exponents equal, but now we would say that x – 1 = 3, therefore x = 4. 2. Take a logarithm of both sides. This methods always works, and for simple situations it is rather quick, but it can also be very cumbersome. Especially if variables are on both sides of the equation.

Example #1 Powers of the Same Base � Solve the equation and confirm your solution with a graph. A. Rewrite both bases to base 3.

Example #1 Powers of the Same Base � Solve the equation and confirm your solution with a graph. B. Rewrite to base 2, remember fractions have negative exponents.

Example #1 continued… Powers of the Same Base � Solve the equation and confirm your solution with a graph. C. Rewrite to base 5.

Example #1 continued… Powers of the Same Base � Solve the equation and confirm your solution with a graph. D. This time the graphs with the intersection method were too difficult to read, so the intercept method was used to verify the two solutions.

Example #2 Logarithms on Both Sides � Solve the equation. Solutions can be confirmed with a graph. Problems like this can be confusing because students will recognize that 2(3) = 6. Unfortunately, that will not work because we have to be able to rewrite both sides to the same base of a power for the first method to be used. Our only choice then is the second method. Notice how common logs and natural logs both work the same way. Choosing which one to use basically comes down to personal preference.

Example #3 � Solve the equation. This type can be very tricky: 1. First take a logarithm of both sides (common or natural). 2. Swing the exponents in front using the properties of logarithms. 3. Distribute. 4. Rewrite the expression so that x terms are on one side of the equation and everything else is on the other side. Remember if you change the side, change the sign. 5. Factor out the GCF of x. 6. Divide.

Example #3 � Solve the equation. Alternatively you can first simplify the answer further by using the properties of logarithms. Typing the answer in the calculator can be difficult:

Example #4 � Solve the equation. 1. Multiply both sides by ex. This will remove the negative exponent on the e. 2. Rewrite the expression equal to 0. 3. Substitute u = ex.

Example #4 � Solve the equation. 1. Use the quadratic formula to solve for u. 2. Substitute ex back in for u. 3. Take the natural logarithm of both sides to eliminate the e on the left side. 4. Only the positive solution from the quadratic formula is used because you can’t take the log of a negative number.

Solving Logarithmic Equations � To solve a logarithmic equation, it is often necessary to condense the expression down to a single logarithm on each side first. This means all the properties of logarithms will come into play. Additionally, constant terms and logarithms cannot be on the same side. � After an individual logarithm is created on each side of the equation, the logarithm can be removed using its inverse, a power of the same base. � Finally, make sure to check for extraneous solutions since the domain of logarithms are limited to numbers greater than 0.

Example #5 � Solve the equation and confirm your solution with a graph. First note that 2(ln x) is the same thing as 2 ln x, which had its coefficient placed up top as an exponent. Secondly, e was chosen as the base of the power because it is the inverse of the natural log. After the logarithms are removed and the expression is set equal to 0, we get a quadratic that cannot factor and will require the Quadratic Formula.

Example #5 � Solve the equation and confirm your solution with a graph. Take for instance the expression: ln(x – 5) If the solutions to an equation with that expression were 4 and 5, even though both answers are positive, when plugged back in we get: When solving this problem we get two solutions. It is important to check both solutions because any value for x (positive or negative) that makes the resulting logarithmic expression negative or 0 will result in no solution. ln(4 – 5) = ln(– 1) ln(5 – 5) = ln(0) Both of these expressions are undefined.

Example #5 � Solve the equation and confirm your solution with a graph. Therefore, the only valid solution is the positive solution: Do not assume this is always the case. A more efficient way of checking will be with the graphing calculator.

Example #5 � Solve the equation and confirm your solution with a graph. A better view: Note: You will have to move the cursor to select each curve when using the intersect method. Secondly, the intersection isn’t showing on this screen, but it is there.

Example #6 � Solve the equation. The trick to this problem is recognizing to the change the addition of two of the same expressions into multiplication by 2. Then both sides are divided by 2 and the problem is solved fairly straightforward.

Example #6 � Solve the equation. Here a second method of solving the same problem is shown but this time two possible answers arrive. After checking for extraneous solutions the same solution is identified.

Example #7 � Solve the equation. The only valid solution is x = 3.

Example #8 Radioactive Decay �It is determined that a mummy has lost 32% of its original carbon-14. When did the person die? Previously, this type of problem was solved by graphing. Now we will solve it algebraically. We also need to know the half-life of carbon-14 which is 5730 years old. If 32% of carbon-14 is lost, then 100% – 32% = 68%

Example #8 Radioactive Decay �It is determined that a mummy has lost 32% of its original carbon-14. When did the person die? We take the logarithm of both sides. Then we swing the exponent out front of the logarithm, and solve for x.

Example #9 Compound Interest �If $8000 is to be invested at 6% per year, compounded monthly, in how many years will the investment be worth $22, 520? Make sure to isolate the base of 1. 005 before taking the log.

Example #10 Population Growth � The population of certain species of rabbit increases exponentially. At the beginning of a 12 -month study, there were 50 rabbits, and at the end of the study there were 164 rabbits. Find the following: A. Find the rate of growth. After solving for a, don’t forget to substitute 1 + r back in for a to find r. Here we see the growth rate is about 10%.

Example #10 Population Growth � The population of certain species of rabbit increases exponentially. At the beginning of a 12 -month study, there were 50 rabbits, and at the end of the study there were 164 rabbits. Find the following: B. Write the function for this population. To keep the function more accurate, we will need to use a growth rate more accurate than 10%. Alternatively, we can use the value for a we found earlier.

Example #10 Population Growth � The population of certain species of rabbit increases exponentially. At the beginning of a 12 -month study, there were 50 rabbits, and at the end of the study there were 164 rabbits. Find the following: C. When will the population reach 1000?

Example #11 Population Growth �The population of a certain bacteria culture at time t hours is given by the function: �How long will it take for the bacteria population to reach 8000? The equation we will need to solve is given by:

Example #11 Population Growth �How long will it take for the bacteria population to reach 8000?