If we want a decimal approximation of the answer, we use a calculator. Sometimes the methods used to solve an equation introduce an extraneous solution , which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.
When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because zero has the unique property that when a product is zero, one or both of the factors must be zero. Keep in mind that we can only apply the logarithm to a positive number.
Always check for extraneous solutions. Skip to main content. Note, when solving an equation involving logarithms, always check to see if the answer is correct or if it is an extraneous solution.
Given an equation containing logarithms, solve it using the one-to-one property. There are two solutions: or The solution is negative, but it checks when substituted into the original equation because the argument of the logarithm functions is still positive. In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa.
We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm. One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life. Figure lists the half-life for several of the more common radioactive substances.
We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time.
We can use the formula for radioactive decay:. How long will it take for ten percent of a gram sample of uranium to decay? Ten percent of grams is grams. If grams decay, the amount of uranium remaining is grams. How long will it take before twenty percent of our gram sample of uranium has decayed?
Access these online resources for additional instruction and practice with exponential and logarithmic equations. Determine first if the equation can be rewritten so that each side uses the same base.
If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.
When does an extraneous solution occur? How can an extraneous solution be recognized? When can the one-to-one property of logarithms be used to solve an equation? When can it not be used? The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base.
For the following exercises, use like bases to solve the exponential equation. For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. For the following exercises, use the definition of a logarithm to solve the equation. For the following exercises, solve each equation for. For the following exercises, solve the equation for if there is a solution.
Then graph both sides of the equation, and observe the point of intersection if it exists to verify the solution. For the following exercises, solve for the indicated value, and graph the situation showing the solution point. An account with an initial deposit of earns annual interest, compounded continuously. How much will the account be worth after 20 years? The formula for measuring sound intensity in decibels is defined by the equation where is the intensity of the sound in watts per square meter and is the lowest level of sound that the average person can hear.
How many decibels are emitted from a jet plane with a sound intensity of watts per square meter? The population of a small town is modeled by the equation where is measured in years.
For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate to 3 decimal places. For the following exercises, use a calculator to solve the equation.
Unless indicated otherwise, round all answers to the nearest ten-thousandth. Atmospheric pressure in pounds per square inch is represented by the formula where is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of pounds per square inch?
Hint : there are feet in a mile. The magnitude M of an earthquake is represented by the equation where is the amount of energy released by the earthquake in joules and is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing joules of energy? Use the definition of a logarithm along with the one-to-one property of logarithms to prove that.
Recall the formula for continually compounding interest, Use the definition of a logarithm along with properties of logarithms to solve the formula for time such that is equal to a single logarithm. Recall the compound interest formula Use the definition of a logarithm along with properties of logarithms to solve the formula for time.
Use the definition of a logarithm along with properties of logarithms to solve the formula for time such that is equal to a single logarithm. Skip to content Exponential and Logarithmic Functions. Learning Objectives In this section, you will: Use like bases to solve exponential equations. Use logarithms to solve exponential equations. Use the definition of a logarithm to solve logarithmic equations.
Use the one-to-one property of logarithms to solve logarithmic equations. Solve applied problems involving exponential and logarithmic equations. Figure 1. Wild rabbits in Australia. In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base.
Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown. Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property. How to: Given an exponential equation with unlike bases, use the one-to-one property to solve it.
Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process? Recall that the range of an exponential function is always positive.
While solving the equation, we may obtain an expression that is undefined. This equation has no solution. The figure below shows that the two graphs do not cross so the left side of the equation is never equal to the right side. Thus the equation has no solution. Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. How to: Solve an exponential equation in which a common base cannot be found.
This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. If we want a decimal approximation of the answer, we use a calculator. We will redo example 5 using this alternate method. The method used in example 5 is good practice using log properties. This alternative approach uses exponent properties instead.
0コメント