Section 3.4 Solving Exponential and Logarithmic Equations

Exponential Equations are equations of the form y = abx Exponential Equations are equations of the form y = abx. When solving, we might be looking for the x-value, the b value or the y-value. First, we’ll review algebraic methods. When solving for b, isolate the b value; then raise both sides of the equation to the reciprocal power of the exponent. Divide both sides by 2 Take the inverse of cube 27 = b3 (27)1/3 = (b3) 1/3 3 = b

When solving for y, solve by performing the indicated operations.   Why did the power become a positive?

When solving for the exponent, rewrite the bases so they have the same base. If the bases are equal, the exponents are equal. Now, solve. Both bases now equal 2 so we can just use the equal exponents.

Solve the following exponential equations: 3 = b y = 25

Solving Exponential Equations by Using the Graphing Calculator Always isolate the variable FIRST!! Graph the function in Y1 Graph the rest of the equation in Y2 Use the intersect function (found with ) to determine the x value

Solve for x to the nearest thousandth: ex=72 Graph Y1 = ex and Y2= 72 Use intersect to give the answer x =4.277

Log Equations are of the form y=logba or y= ln x where the base is e. Y= log216 4=log3x 3= logx1000 To solve for x, we need to undo the log format by rewriting in exponential form. 2y=16 34=x x3=1000 Now we use the exponential rules to solve. 2y=24 81 = x x3 = 103 y = 4 x = 10

ex = 72 ln ex = ln 72 Solve algebraically: Rewrite as an ln equation: Since ln ex means the exponent of ln ex, just use x:   Note: when an equation is written in terms of e, you MUST use natural logs. Otherwise you may use log or ln at will.

Solve for x algebraically: 2x = 14 Take the ln or log of each side and solve.

Solve for x: 2x=14 graphically. Graph Y1 = 2 x and Y2= 14 Use intersect to approximate the answer X=3.807

Solve lnx = 3 to the nearest tenth.

Solve for x: X= 5 X=-2

More Involved Equations Sometimes log or ln equations require a few more steps to “clean them up” before we can simply “take the log” or “undo the log”. Before you can take the ln, you need to isolate it! 5 + 2ln x = 4 2ln x = -1 ln x = -0.5 e ln x = e-0.5 x= 0.60653…

To solve graphically, you still must isolate the ln expression To solve graphically, you still must isolate the ln expression. 2ln x = -1 Graph as y1 = 2ln x y2= -1

Solve ln 3x = 2   Solve 2 log5 3x = 4  

ln x = x2 – 2 ln (x) – x2 + 2 = 0 Solve using the graphing calculator: This problem has 2 solutions. Check them both.

More practice

Solving Logarithmic Equations Algebraically Using Laws of Logarithms When an equation contains the word log or ln, we need to eliminate it to solve the equation so first we apply the laws of logarithms to “undo” the addition by changing to multiplication, “undo” subtraction by changing it to division, and “undo” powers by changing them to multiplication.. Solve: log 2(4x+10) – log2(x+1) = 3

Log 2(4x+10) – log2(x+1) = 3 Apply Quotient Rule. Definition of Logarithm Cross multiply and solve

Solve:

Practice, Practice, Practice Pg 253 Vocabulary Check 1-3 Pg 253 Exercises 1, 3, 5, 9-23 every other odd, 25-105 multiples of 5, 107-115 odds