4 Exponential and Logarithmic Functions

4.6 Exponential and Logarithmic Equations

Objectives Use Like Bases to Solve Exponential Equations Use Logarithms to Solve Exponential Equations Solve Logarithmic Equations Solve Carbon-14 Dating Problems Solve Population Growth Problems

Exponential Equations An exponential equation is an equation with a variable in one of its exponents. Some examples of exponential equations are 3x = 5 e2x = 7 6x–3 = 2x 32x+1 – 10(3x) + 3 = 0

Logarithmic Equations A logarithmic equation is an equation with logarithmic expressions that contain a variable. Some examples of logarithmic equations are log 2x = 25 ln x – ln(x – 12) = 24 log x = log (1/x) + 4

1. Use Like Bases to Solve Exponential Equations

Property of Exponents If bx = by, then x = y. That is, equal quantities with like bases have equal exponents.

Example 1 Solve: 4x+3 = 82x

Example 1 – Solution We will use like bases to solve the exponential equation.

2. Use Logarithms to Solve Exponential Equations

Example 4 Solve the exponential equation: 3x = 5

Example 4 – Solution Since logarithms of equal numbers are equal, we can take the common logarithm of each side of the equation. We then can use the power rule and move the variable x from its position as an exponent to a position as a coefficient and solve the equation.

Example 4 – Solution To four decimal places, x = 1.4650.

Comment A careless reading of previous example leads to a common error. The right side of the equation calls for a division, not a subtraction. It is the expression log (5/3) that means log 5 – log 3.

3. Solve Logarithmic Equations

Property of Logarithms If logb x = logb y, then x = y. That is, logarithms of equal numbers are equal.

Example 7 Solve: log5 (3x + 2) = log5 (2x – 3)

Example 7 – Solution We will use the property of logarithms stated above to solve the equation.

4. Solve Carbon-14 Dating Problems

Carbon-14 When a living organism dies, the oxygen/carbon dioxide cycle common to all living things ceases. Then carbon-14, a radioactive isotope with a half-life of 5,700 years, is no longer absorbed. By measuring the amount of carbon-14 present in ancient objects, archaeologists can estimate the object’s age.

Radioactive Decay Model The amount A of radioactive material present at time t is given by the model A = A02–t/h where A0 is the amount present initially and h is the half-life of the material.

Example 10 An archeologist finds a wooden statue in the tomb of all ancient Egyptian ruler. If the statue contains two-thirds of its original carbon-14 content, how old is it?

Example 10 – Solution To find the time t when A = (2/3)A0, we substitute 2A0/3 for A and 5,700 for h in the radioactive decay formula and solve for t:

Example 10 – Solution The wooden statue is approximately 3,300 years old.

5. Solve Population Growth Problems

The Malthusian Growth Model When there is sufficient food and space, populations of living organisms tend to increase exponentially according to the Malthusian growth model P = P0ekt where P0 is the initial population at t = 0 and k depends on the rate of growth.

Example 11 Streptococcus bacteria in a laboratory culture increased from an initial population of 500 to 1,500 in 3 hours. Find the time it will take for the population to reach 10,000.

Example 11 – Step 1 Substitute 1,500 for P, 500 for P0, and 3 for t into the Malthusian growth model and find k.

Example 11 – Step 2 Substitute 10,000 for P, 500 for P0, and the value of k found in Step 1 into the model and use logarithms to solve for t. The culture will reach 10,000 bacteria in a little more than 8 hours.