Logarithmic Functions Topic 3: Solving Exponential Equations Using Logarithms

I can determine the solution of an exponential equation by using logarithms. I can solve problems that involve the application of exponential equations to loans, mortgages, and investments. I can solve problems that involve logarithmic scales, such as the Richter scale and the pH scale.

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You Should Notice… There are two ways that you can solve an exponential function algebraically: Write each side as a power with the same base, and then equate the exponents and solve for the variable. You can take the logarithm of both sides and use log laws to isolate and solve for the variable.

Information In the Exponential Functions unit, we saw that an exponential equation is an equation that has a variable in an exponent. There are three ways to solve exponential equations. ① Graphically Graph the left-hand side of the equation in Y 1. Graph the right-hand side of the equation in Y 2. The x-value of the intersection is the solution to the equation. ② Equating the exponents when the bases are the same If the bases are already the same, equate the exponents and solve for the variable. If the bases are different, but they can be rewritten with the same base, then use the exponent laws, equate the exponents, and solve for the variable.

Information ③ Taking the logarithm of both sides of the equation If the bases cannot be rewritten using a common base, then use logarithms to solve the equation. If two expressions are equal, then taking the common logarithm of each expression maintains the equality of the expressions. If M = N, then, (where M>0, N>0, b>0, b≠1, and M and N are real numbers). Then use the logarithm laws to solve for the variable.

Example 1 Solve the following exponential equation using the methods listed below. a) change of baseb) graphically Selecting a method for solving an exponential equation Write each side of the equation with a base 3 power. Find the intersection point (2 nd Trace 5:intersect). X = -5

Example 1 c) using logs Selecting a method for solving an exponential equation Take the common logarithm (base 10) of each side. Expand the logarithms (bring any exponents down). Use algebra to isolate the x-value.

Example 2 Solve the following exponential equations by taking the logarithm of both sides of the equation, rounded to the nearest hundredth. a) Solving an exponential equation by taking the logarithm of both sides Take the common logarithm (base 10) of each side. Expand the logarithms (bring any exponents down). Use algebra to isolate the x-value.

Example 2 b) Take the common logarithm (base 10) of each side. Expand the logarithms (bring any exponents down). Use algebra to isolate the x-value. Before we start this one, we move the constant to the other side of the equals sign.

Example 3 The half-life equation is, where A represents the amount of the substance remaining, A o represents the initial amount of the substance, t represents the time, and h represents the time at which only half of the substance remains. Jahmal laboratory received a shipment of 500 g of radioactive radon-222 which has a half-life of 3.8 days. By the time it was needed, only g remained. Algebraically determine the length of time (to the nearest day) that Jahmal laboratory had the radioactive radon-222 before needing it. Using logarithms to solve a half-life problem 20 days. Divide both sides by 500. Change to log form. Multiply both sides by 3.8.

Example 4 Yvonne has a balance of $3 215 in her savings account. This account pays interest at a rate of 2.4%/a, compounded annually. The compound interest formula is where A represents the future value, P represents the principal, i represents the interest rate applied each compounding period, and n represents the number of compounding periods. Determine how long it will take for Yvonne’s balance to reach $5 000, to the nearest tenth of a year. Solving a compound interest problem using logarithms

Example 4 Yvonne has a balance of $3 215 in her savings account. This account pays interest at a rate of 2.4%/a, compounded annually. Determine how long it will take for Yvonne’s balance to reach $5 000, to the nearest tenth of a year. Principal (P) = 3215 Accumulated amount (A) = years

In the Exponential Functions unit, we saw that an exponential equation is an equation that has a variable in an exponent. There are three ways to solve exponential equations. ① Graphically Graph the left-hand side of the equation in Y 1. Graph the right-hand side of the equation in Y 2. The x-value of the intersection is the solution to the equation. ② Equating the exponents when the bases are the same If the bases are already the same, equate the exponents and solve for the variable. If the bases are different, but they can be rewritten with the same base, then use the exponent laws, equate the exponents, and solve for the variable. Need to Know

③ Taking the logarithm of both sides of the equation If the bases cannot be rewritten using a common base, then use logarithms to solve the equation. If two expressions are equal, then taking the common logarithm of each expression maintains the equality of the expressions. If M = N, then Then use the logarithm laws to solve for the variable. Need to Know You’re ready! Try the homework from this section.