College Algebra Chapter 4 Exponential and Logarithmic Functions Section 4.5 Exponential and Logarithmic Equations and Applications Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Solve Exponential Equations Solve Logarithmic Equations Use Exponential and Logarithmic Equations in Applications

Concept 1 Solve Exponential Equations

Solve Exponential Equations Some exponential equations can be solved using the equivalence property of exponential expressions: If b, x, and y are real numbers with b > 0 and b ≠ 1, then Most exponential equations cannot be rewritten to have matching bases. To solve these equations, isolate the exponential expression, take a logarithm of both sides, and then apply the power property of logarithms.

Example 1 Solve Solution:

Example 2 Solve Solution:

Example 3 Solve Solution:

Skill Practice 1 Solve.

Example 4 Solve Solution:

Example 5 Solve Solution:

Skill Practice 2 Solve.

Example 6 Solve Solution:

Skill Practice 3 Solve.

Example 7 Solve Solution:

Skill Practice 4 Solve.

Example 8

Skill Practice 5 Solve.

Concept 2 Solve Logarithmic Equations

Solve Logarithmic Equations (1 of 2) Note the difference between these two equations: In the first equation, all terms have a logarithm. The second equation is a mix of logarithmic and constant terms. These two equations require different styles of solution.

Solve Logarithmic Equations (2 of 2) For solving equations of the first type, we will use the equivalence property of logarithmic expressions. If b, x, and y are real numbers with b > 0 and b ≠ 1, then To solve equations that are a mix of logarithms and constants, collect all logarithms together, then rewrite the equation in exponential form.

Example 9 Solve Solution:

Example 10 Solve log (x - 7) = log (5 - x) Solution: x – 7 = 5 - x 2x = 12 x = 6 log (5 - x) = log (5 - 6) = log (-1) undefined no solution

Skill Practice 6 Solve.

Example 11 Solve log (3x - 4) – log (x + 1) = log 2 Solution:

Example 12 Solve ln (x + 3) + ln x = ln 10 Solution:

Example 13 Solve Solution:

Skill Practice 7 Solve. ln x + ln (x - 8) = ln (x - 20)

Example 14 Solve Solution:

Example 15 Solve Solution:

Example 16 Solve log (x + 15) = 2 – log x Solution: log (x + 15) = 2 – log x log (x + 15) + log x = 2 log x(x + 15) = 2 log base is 10

Example 17 Solve Solution:

Skill Practice 8 Solve.

Skill Practice 9 Solve. log (t - 18) = 1.4

Skill Practice 10 Solve.

Concept 3 Use Exponential and Logarithmic Equations in Applications

Example 18 If $20,000 is invested in an account earning 2.5% interest compounded continuously, determine how long it will take the money to grow to $45,000. Round to the nearest year. Use the model Solution: A = 45000, P = 20000, r = 0.025

Skill Practice 11 Determine how long it will take $8000 compounded monthly at 6% to double. Round to 1 decimal place.

Example 19 (1 of 2) The following formula relates the energy E (in joules) released by an earthquake of magnitude M. log E = 5.24 + 1.44M (M > 5) What is the energy released by a magnitude 6 and a magnitude 7 earthquake? Solution:

Example 19 (2 of 2) As a comparison, a 1 megaton nuclear weapon would have an energy release of joules and the eruption of the volcano Krakatoa in 1883 produced an energy release of

Skill Practice 12 Find the intensity of sound from a leaf blower if the decibel level is 115 dB. Is the intensity of sound from a leaf blower above the threshold for pain?