7.6 solving logarithmic equations

Exponential & Logarithmic Equations

• Exponential Equations with Like Bases

• Exponential Equations with Different Bases

• Logarithmic Equations

Exponential Equations with Like Bases

In an Exponential Equation, the variable is in the exponent. There may be one exponential term or more than one, like…

If you can isolate terms so that the equation can be written as two expressions with the same base, as in the equations above, then the solution is simple.

32x1 5 4 or 3x1 9x 2


Example #1 - One exponential expression.

32x1 5 4

32x1 9

32x1 32

1. Isolate the exponential expression and rewrite the constant in terms of the same base.

2x 1 2

2x 1

x 1

2

2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.


Example #2 - Two exponential expressions.

3x1 9x 2

3x1 32 x 2

3x1 32x 4

1. Isolate the exponential expressions on either side of the =. We then rewrite the 2nd expression in terms of the same base as the first.

2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

x 1 2x 4

x 5

Change-of-Base Formula

The base you change to can be any base so generally we’ll want to change to a base so we can use our calculator. That would be either base 10 or base e.

LOG

“common” log base 10

LN

“natural” log base e

a

M

log

log

a

M

ln

ln

Example for TI-83

If we generalize the process we just did we come up with the:

a

MM

b

ba log

loglog

Use the Change-of-Base Formula and a calculator to approximate the logarithm. Round your answer to three decimal places.

16log3

Since 32 = 9 and 33 = 27, our answer of what exponent to put on 3 to get it to equal 16 will be something between 2 and 3.

3ln

16ln16log3

put in calculator

524.2

Exponential Equations with Different Bases

The Exponential Equations below contain exponential expressions whose bases cannot be rewritten as the same rational number.

The solutions are irrational numbers, we will need to use a log function to evaluate them.

32x1 5 11 or 3x1 4 x 2



1. Isolate the exponential expression.

3. Use the log rule that lets you rewrite the exponent as a multiplier.

32x1 5 11

32x1 162. Take the log (log or ln) of both sides of the equation.ln 32x1 ln 16

(2x 1)ln 3 ln16



4. Isolate the variable.

2x 1 ln16

ln 3

2x ln16

ln 3 1

x ln16

2 ln 3

1

2x 0.762

(2x 1)ln 3 ln16



1. The exponential expressions are already isolated.

3. Use the log rule that lets you rewrite the exponent as a multiplier on each side..

2. Take the log (log or ln) of both sides of the equation.

3x1 4 x 2

ln 3x1 ln 4 x 2 (x 1)ln 3 (x 2)ln 4



4. To isolate the variable, we need to combine the ‘x’ terms, then factor out the ‘x’ and divide.

x ln 3 ln 3 x ln 4 2 ln 4

x ln 3 x ln 4 ln 3 2 ln 4

x(ln 3 ln 4) (ln 3 2 ln 4)

x (ln 3 2 ln 4)

ln 3 ln 4

x 13.457

Logarithmic Equations

In a Logarithmic Equation, the variable can be inside the log function or inside the base of the log. There may be one log term or more than one. For example …

log4 2x 1 3 5

ln x ln(2x 1) 1

logx 3 2


Example 1 - Variable inside the log function.

log4 2x 1 3 5

log4 2x 1 2

42 2x 1

16 2x 1

2x 17

x 8.5

1. Isolate the log expression.

2. Rewrite the log equation as an exponential equation and solve for ‘x’.


Example 2 - Variable inside the log function, two log expressions.

ln x ln(2x 1) 1

lnx

2x 1

1

e1 x

2x 1e(2x 1) x

2ex ex2ex x ex(2e 1) e

x e

2e 1

1. To isolate the log expression, we 1st must use the log property to combine a difference of logs.

2. Rewrite the log equation as an exponential equation (here, the base is ‘e’).

3. To solve for ‘x’ we must distribute the ‘e’ and then collect the ‘x’ terms together and factor out the ‘x’ and divide.

x 0.613


Example 3 - Variable inside the base of the log.logx 3 2

x 2 3

x 2 12 3 1

2

x 1

3

x 3

3

1. Rewrite the log equation as an exponential equation.

2. Solve the exponential equation.

7.6 solving logarithmic equations

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